Transcript
Page 1: Introductory physics students' conceptions of algebraic signs used in

Examensarbete  C,  15  hp     Juni  2014  Uppsala  universitet    

Introductory physics students’ conceptions of algebraic signs used in kinematics problem solving Moa Eriksson                      Supervisor: Cedric Linder Subject reader: John Airey Divisions of Physics Education Research, Department of Physics and Astronomy

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Abstract The  ways  that  physics  students’  conceptualize  –  the  way  they  experience  –  the  use  of  algebraic  signs   in   vector-­‐kinematics  has  not  been   extensively   studied.  The  most   comprehensive  of   these  few  studies  was  carried  out  in  South  Africa  15  years  ago.  This  study  found  that  the  variation  in  the   ways   that   students   experience   the   use   of   algebraic   signs   could   be   characterized   by   five  qualitatively  different  categories.  The  consistency  of   the  nature  of   this  experience  across  either  the  same  or  different  educational  settings  has  never  given  further  consideration.  This  project  sets  out  to  do  this  using  two  educational  settings;  one  similar  to  the  original  South  African  one,  and  one   at   the   natural   science   preparatory   programme   known   as  basåret   at   Uppsala   University   in  Sweden.    

The  study  was  carried  out  under   the  auspices  of   the  Division  of  Physics  Education  Research  at  the  Department  of  Physics  and  Astronomy  at  Uppsala  University   in   collaboration  with  Nadaraj  Govender,  University  of  KwaZulu-­‐Natal,  who  performed  the  original  study  while  completing  his  PhD  at  the  University  of  the  Western  Cape,  South  Africa.    

This  study  is  situated  in  the  kinematics  section  of  introductory  physics  with  participants  drawn  from   the   natural   science   preparatory   programme   at   Uppsala   University   and   physical   science  preservice   teachers’   programme   at   the   University   of   KwaZulu-­‐Natal,   South   Africa.   The  participating   students   completed   a   specially   designed   questionnaire   on   the   use   of   signs   in  kinematics  problem  solving.  A  sub-­‐group  of  these  students  was  also  purposefully  selected  to  take  part  in  semi-­‐structured  interviews  that  aimed  at  further  exploring  their  experiences  of  algebraic  signs.  The  students’  descriptions  and  answers  were  categorized  using  Nadaraj  Govender’s  set  of  categories,   which   had   been   constructed   using   the   phenomenographic   research   approach.   This  approach  is  designed  to  enable  finding  the  variation  of  ways  people  experience  a  phenomenon.  The  process  of  sorting  the  data  was  grounded  in  this  phenomenographic  perspective.  From  this  categorization   it   was   possible   to   identify   four   of   the   original   five   categories   amongst   the  participating  students.    

The   results   suggest   that   these   four   categories   remain   educationally   relevant   today   even   if   the  context   is   not   the   same   as   the   one   for   the   original   findings.   Although   one   of   the   original   five  categories   was   not   found,   the   analysis   cannot   be   taken   to   definitely   eliminate   this   from   the  original  outcome  space   of   results.  A  more  extensive   study  would  be  needed   for   this   and   thus  a  proposal  is  made  that  further  studies  be  undertaken  around  this  issue.  

The   study   ends   by   suggesting   that   physics   teachers   at   the   introductory   level   need   to   obtain   a  broader   understanding   of   their   students’   difficulties   and   develop   their   teaching   to   better   deal  with  the  challenges  that  become  more  visible  in  this  broader  understanding.    

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Sammanfattning På   vilka   sätt   fysikstudenter   föreställer   sig   och   förstår   användandet   av   algebraiska   tecken   i  vektorkinematik  har   endast   studerats   i  mindre  utsträckning.  Den  mest   omfattande   av  dessa   få  studier   genomfördes   i   Sydafrika   för  15  år   sedan.  Denna   studie  upptäckte  att   variationen  av  de  sätt  studenter  upplever  användandet  av  algebraiska  tecken  på  kunde  karaktäriseras  genom  fem  kvalitativt   olika   kategorier.   Hur   solida   dessa   upplevelser   är   i   en   liknande   eller   helt   annan  utbildningsmiljö  har  däremot  inte  studerats  vidare.  Detta  projekt  ämnar  till  att  göra  detta  genom  att   använda   två   olika   studentgrupper;   en   liknande  den  ursprungliga   gruppen   i   Sydafrika,   samt  det  tekniskt-­‐naturvetenskapliga  basåret  vid  Uppsala  universitet,  Sverige.    

Studien   har   genomförts   med   stöd   från   avdelningen   för   fysikens   didaktik   vid   institutionen   för  fysik   och   astronomi   vid   Uppsala   universitet   i   samarbete  med  Nadaraj   Govender,   University   of  KwaZulu-­‐Natal,   Sydafrika,   som   genomförde   den   ursprungliga   studien   under   sin  doktorandutbildning  vid  University  of  the  Westen  Cape,  Sydafrika.  

Denna  studie  är  begränsad  till  den  del  av  den  grundläggande   fysiken  som  behandlar  kinematik  och   innefattade   deltagare   från   det   tekniskt-­‐naturvetenskapliga   basåret   vid  Uppsala   universitet  samt   tredje   års   studenter   vid   physical   science   preservice   teachers’   programme,   University   of  KwaZulu-­‐Natal,   Sydafrika.   De   deltagande   studenterna   genomförde   ett   specialdesignat  frågeformulär  kring  användandet  av  algebraiska  tecken  för  att  lösa  kinematiska  problem.  En  del  av  dessa  studenter  valdes  sedan  ut  för  att  delta   i  semi-­‐strukturerade  intervjuer  som  syftade  till  att  vidare  utforska  deras  upplevelser  kring  algebraiska   tecken.  Studenternas  beskrivningar  och  svar  kategoriserades  med  hjälp  av  Nadaraj  Govenders  fem  kategorier  som  tagits  fram  genom  ett  fenomenografiskt   tillvägagångssätt.   Detta   tillvägagångssätt   är   framtaget   för   att   kunna   hitta  variationen   av   hur   människor   upplever   ett   fenomen.   Sorteringsprocessen   grundades   i   detta  fenomenografiska  perspektiv.  Från  denna  kategorisering  var  det  möjligt  att  identifiera  fyra  av  de  fem  ursprungliga  kategorierna  bland  de  deltagande  studenterna.    

Fyra  av  de   fem  ursprungliga  kategorierna  som  föreslagits  av  Govender  återfanns  genom  denna  studie  varför  dessa  kategorier  föreslås  förbli  relevanta  idag  även  om  utbildningsmiljön  skiljer  sig  från   den   ursprungliga.   Trots   att   den   femte   kategorin   inte   hittades   kan   denna   inte   definitivt  exkluderas  från  det  outcome  space  som  beskriver  studenters  upplevelser  för  algebraiska  tecken.  Det  föreslås  att  vidare  studier  undersöker  förekomsten  av  denna  kategori.    

Studien  avslutas  med  att  föreslå  att  fysik  lärare  på  grundnivå  behöver  få  en  bättre  förståelse  för  sina  studenters  svårigheter  samt  att  de  behöver  utveckla  sin  undervisning   för  att  bättre  kunna  hantera  dessa  svårigheter  och  på  så  sätt  göra  undervisningen  mer  anpassad   för  mångfalden  av  studenterna.    

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Acknowledgements To  my  mother  and   father   for  believing   in  me  and  encouraging  me   to   fulfill  my   studies  and  doing  what  I  believe  in.      

I  would  like  to  express  my  deepest  appreciations  to  my  supervising  professor,  Cedric  Linder,  for  his   continuous   interest,   guidance   and   engagement   in   the   development   of   this   study.   All   your  encouragement   throughout   this   project   has   been   invaluable   and   has   helped   me   complete   my  thesis!  I  would  also  like  to  thank  Dr.  Nadaraj  Govender  for  his  feedback  on  my  study  design,  his  encouragement   and   for   his   valuable   support   as   my   external   collaborator   during   the   project.  Special   thanks   goes   to   Jonas   Forsman   for   his   interest   and   guidance   from   the   very   start   of  my  project,  which  helped  me  to  quickly  progress.    

I  would   further   like   to   thank  all  members  of   the  Division  of  Physics  Education  Research  at   the  Department  of  Physics  and  Astronomy  at  Uppsala  University,  especially  Anne  Linder,  for  making  me  feel  truly  welcome  as  a  part  of  your  group.      

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Table of contents  

1   INTRODUCTION  ................................................................................................  1  

1.1   Problem  setting  .....................................................................................................................................  1  1.1.1  Research  question  ...............................................................................................................................................  2  

1.2   Object  ........................................................................................................................................................  2  

1.3   Goal  ...........................................................................................................................................................  2  

2   BACKGROUND  ..................................................................................................  3  

3   METHODOLOGY  ...............................................................................................  5  

3.1   Theory  ......................................................................................................................................................  5  

3.2   Method  .....................................................................................................................................................  6  3.2.1  Ethics  ........................................................................................................................................................................  6  3.2.2  Validity  and  reliability  .......................................................................................................................................  7  3.2.3  Questionnaire  ........................................................................................................................................................  7  3.2.3.1   Design  .........................................................................................................................................................  8  3.2.3.2   Pilot  studies  .............................................................................................................................................  9  3.2.3.3   Translation  ...............................................................................................................................................  9  

3.2.4  Interviews  ...............................................................................................................................................................  9  3.2.4.1   Design  .......................................................................................................................................................  10  3.2.4.2   Pilot  interview  ......................................................................................................................................  10  3.2.4.3   Transcriptions  ......................................................................................................................................  10  

3.2.5  Analysis  ..................................................................................................................................................................  10  

4   RESULTS  .........................................................................................................  11  

4.1   Summary  of  findings  ........................................................................................................................  12  

4.2   Evaluation  ............................................................................................................................................  13  

4.3   Description  of  categories  ................................................................................................................  13  

5   DISCUSSION  ....................................................................................................  17  

5.1   Identified  challenges  ........................................................................................................................  18  

6   RECOMMENDATIONS  .....................................................................................  20  

7   CONCLUSIONS  ................................................................................................  20  

REFERENCES  .........................................................................................................  22  

APPENDIXES  .........................................................................................................  24  

Appendix  1  ......................................................................................................................................................  24  

Appendix  2  ......................................................................................................................................................  28  

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1 Introduction Mathematically,   when   using   vector   notation   the   choice   of   an   appropriate   sign   is   largely   an  arbitrary  decision,  however,  in  physics  problem  solving,  a  set  of  conventions  that  have  a  strong  conceptual   base   usually   guide   their   use.   For   example,   in   problems   involving   the   direction   an  object  is  moving  when  restricted  to  one  dimension,  a  range  of  possible  temperatures,  the  sign  of  an   electric   charge,   and   the   gain   or   loss   of   energy   from   a   specified   system.   All   of   these   use  algebraic   signs,   which   have   distinct   conceptual  meanings   as   a   function   of   their   context.   Thus,  getting   to   appropriately   understand   how   to   conceptualize   the   use   of   algebraic   signs   across  contexts  and  phenomena  is  an  important  aspect  of  learning  in  introductory  physics.  

From  an  early  stage,  throughout  school,  students  meet  the  use  of  the  algebraic  signs  “plus”  and  “minus”   across   different   contexts.   However,   these   appear   to   become   understood   procedurally  without   an   accompanying   comprehensive   and   appropriate   conceptual   anchoring.   Such  procedural   knowledge   has   limited   sustainability   in   introductory   university   physics   and   little  value  for  the  study  of  more  advanced  physics.  As  students  go  from  one  problem  solving  context  to  another  a   lack  of   conceptual  grounding  can  easily  generate  challenges  both   for   introductory  problem  solving  and  for  more  advanced  problem  solving.  For  example,  at  the  introductory  level  -­‐6   may   be   considered   to   be   mathematically   smaller   than   -­‐4,   yet   a   velocity   of   -­‐6   m/s   may   be  considered  to  be  larger  than  a  velocity  of  -­‐4m/s,  and  making  sense  of  the  sign  convention  for  the  basic  Dirac  equation  using  applications  of  the  Minkowski  metric  can  be  challenging.    

When   dealing   with   vectors   in   introductory   university   physics   the   issue   of   appropriately  understanding   the   signs   that  get  used  becomes  more  problematic.  For  example,   students  often  have  not  been  introduced  to  the  use  of  vectors  in  more  than  one  way  and  they  get  to  think  about  component  vectors  in  vector  terms  rather  than  scalar  terms.  When  this  happens  the  sign  used  is  often  taken  to  denote  direction  and  basic  scalar  additions  become  convoluted  with  perceptions  of  vector  direction.  A  further  example  is  that  students  may  want  the  sign  conventions  to  continue  to  fit   the   conceptualization   of   the   meaning   that   they   have   already   constructed.   For   example,   a  negative   velocity  may   be   taken   to  mean   “slowing   down”   rather   than   designating   the   labelling  being   used   from   the   establishment   of   a   coordinate   system   and   the   direction   within   one  dimension  of  that  coordinate  system.  

1.1 Problem setting This   project   builds   upon   previous   research   that   will   be   further   described   in   the   Background  section  below.  Only   a   few   studies   have   investigated   students’   use   of   algebraic   signs   in   physics  problem  solving  (for  example,  Viennot  2004;  Hayes  &  Wittmann  2010).  The  most  comprehensive  of  these  are  the  studies  carried  out  by  Govender  (1999;  2007).  Govender  (1999)  used  his  analysis  to   generate   qualitatively   different   categories   of   description   of   the   experience   of   using   signs   in  kinematics   problem   solving.   Later   Govender   (2007)   expanded   upon   these   results.   In   both  studies,   Govender   drew   on   a   phenomenography   framing   (Marton   &   Booth   1997)   to   provide  details   of   the  outcome   space   for   the  qualitatively  different  ways   students   experience   signing   in  introductory  physics.  This  outcome  space  consisted  of  five  different  categories  each  reflecting  the  variation   in   the   ways   that   signs   get   conceptualized   in   introductory   kinematics   physics.   The  outcome  space  together  with  Govender’s  original  study  can  be  found  in  appendix  2.      

Govender’s  studies  were  carried  out  in  a  very  different  socio-­‐economic  setting  to  that  of  Sweden  but  with  a  programme  that  has  many  similarities  to  the  natural  science  preparatory  programme  in  Sweden  known  as  basåret.  The  data  set  that  Govender  used  is  now  nearly  15  years  old  and  the  changing  educational  experience  in  physics  internationally  (Henderson,  Dancy  &  Niewiadomska-­‐Bugaj  2012),  could  reasonably  be  expected  to  affect   the  current  applicability  of   the  results   that  Govender  obtained.  Using  this  as  a  starting  point,  the  following  research  question  was  developed.  

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1.1.1 Research question For  Swedish  and  South  African  student  populations  who  are  enrolled   in  a  physics   introductory  course  which   has   a   similar   approach   to   that   of   the   natural   science   preparatory   programme   in  Sweden  known  as  basåret,  the  specific  research  question  for  this  study  is  the  following:  

• How  relevant  are  Govender’s  (1999;  2007)  results  for  the  variation  in  ways  that  the  use  of  algebraic   signs  are  experienced   for  one-­‐dimensional  kinematics  problem  solving   for  students   in  the  natural  science  preparatory  programme  at  Uppsala  University,  Sweden,  and  the  third  year  physical  science  preservice  teachers’  programme  at  the  University  of  KwaZulu-­‐Natal,  South  Africa?  

1.2 Object To   achieve   a   good  quality   of   physics   education,   teachers  must   be   aware   of   the   challenges   that  students   experience   when   being   introduced   to   the   use   of   vectors.   If   teachers   can   attain   this  awareness  they  will  be  able  to,   in  a  better  way,  address  the  diversity  of   the  student  group.  The  study   performed   by   Govender   (1999)   shows   in   what   different   ways   introductory   physics  students  in  KwaZulu-­‐Natal,  South  Africa,  experience  the  use  of  algebraic  signs  in  one-­‐dimensional  kinematics  problem  solving.  However,  no   further   research  has  been   carried  out   to   study   if   the  results  can  be  generalized  for  use  in  explaining  the  understanding  of  signs  among,  for  example,  introductory  physics  students  in  Uppsala,  Sweden.    

Thus,   the   object   of   this   project   is   to   study   the   different   ways   students   in   the   natural   science  preparatory   programme   in   Uppsala,   Sweden,   and   physical   science   preservice   teachers   at   the  University  of  KwaZulu-­‐Natal,   South  Africa,   experience   the  sign  conventions  used   for  describing  three   fundamental  concepts   in  Newtonian  mechanics   (displacement,  velocity  and  acceleration).  These   experiences   will   be   categorised   using   Govender’s   (2007)   categories   in   order   to,   in   the  future,  make  it  easier  for  teachers  to  address  the  variation  of  physics  students’  experiences  when  learning  algebraic  signs.    

Since  Govender’s  (1999)  study,  there  has  been  no  further  research  in  the  particular  area  of  trying  to   categorise   students’   different   ways   of   understanding   the   sign   conventions   used   in   vector  kinematics.  Many  studies  have  been  made   investigating  students’  understanding  of  vectors  and  sign   conventions   in   physics   (see,   for   example,   Aguirre   1988;   Hayes   &   Wittmann   2010;  McDermott   1984),   however   no   one   has   further   tried   to   generalise   Govender’s   discovered  categories  in  the  area  of  kinematics.  Thus,  it  is  of  great  interest  to  perform  a  study  where  the  five  qualitatively  different  ways  students  experience   the  use  of   signs   in  vector  kinematics   found  by  Govender  (2007)  are  reviewed  to  investigate  their  generalizability.    

1.3 Goal The   goal   of   this   project   is   to   investigate   how   introductory   level   physics   students   at   Uppsala  University   conceptualize   the   way   they   use   signs   in   physical   problem   solving   in   the   area   of  kinematics,  and  to  use  this  analysis  to  compare  and  contrast  results  obtained  at  the  University  of  KwaZulu-­‐Natal   in  South  Africa.  This   comparison  will  be  used   to   identify  generalizable   learning  challenges   in   this   area   in   order   to   inform   the   development   of   associated   physics   education.  Knowing   the   nature   of   the   identified   generalizability   would   provide   a   powerful   platform   to  inform  the  teaching  and  learning  of  kinematics  in  ways  that  better  accommodate  the  diversity  of  student  population  found  in  the  introductory  level  of  physics  education  in  Sweden  today.  

 

 

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2 Background Different   challenges   that   students   face  when   learning   physics   have   been   investigated   in  many  studies  over  a  long  period  of  time.  For  example,  research  has  shown  that  students  have  difficulty  understanding   negative   velocities   and   how   to   link   them   to   a   physical   situation   (Goldberg   &  Anderson   1989;   Testa,   Monroy   &   Sassi   2002).   When   presented   with   a   velocity-­‐time   diagram  many  students  failed  to  recognise  the  motion  of  an  object  when  the  diagram  showed  a  negative  velocity  and  could  not  with  certainty  draw  a  diagram  of  the  motion  by  themselves.  This  difficulty,  understanding  the  meaning  of  negative  velocity,  could  stem  from  the  fact  that  students  often  are  only  exposed  to  vectors  in  one  way.  For  example,  students  are  used  to  describing  vectors  through  equations  and  drawing  vector  arrows  in  a  coordinate  system,  where  a  negative  velocity  usually  is  directed  to  the  left.  However,  by  just  drawing  vector  arrows,  students  fail  to  experience  the  link  between  the  vector  and  a  physical  situation.    

It   has   further   been   argued   in   the   literature   that   the   difficulties   in   understanding   the   use   of  algebraic  signs  that  are  commonly  used  in  kinematics  problems  can  often  be  traced  back  to  the  misuse  of  the  correct  signs  in  physics  textbooks  (for  example,  see  Brunt  1998).  When  deciding  on  a  sign  for  a  quantity  in  physics  problem  solving  there  are  some  rules,  which  have  to  be  followed,  that   textbook  writers   frequently  do  not   follow  or  perhaps  are  not  aware  of   (Brunt  1998).  Such  lack  of  coherence  and  systemization  can  easily  generate  ambiguities  in  what  meanings  are  being  signified  in  the  ways  the  signs  get  used.  In  order  to  avoid  such  problems  emerging,  Brunt  (1998;  242)   proposes   that   teachers   of   physics   incorporate   two   simple   guidelines   into   their   teaching  practice  when  teaching  at  the  introductory  level:  (1)  when  deriving  an  equation  always  “draw  a  diagram  with   all   variables   in   a   chosen   positive   direction”,   and   (2)   to   “never   substitute   a   sign  unless  substituting  a  number,  or  its  algebraic  equivalence”.    

The  first  of  these  two  guidelines  proposes  that  one  should  first  decide  on  a  positive  direction  in  order  to  draw  an  appropriate  coordinate  system  to  situate  the  needed  diagram.  This  is  verified  in  a  study  (Viennot  2004)  that  argues  that  one  often  has  to  decide  on  an  axis  of  reference  in  order  to  assign   the   appropriate   sign   to   a   quantity.  However,   this  does  not  mean  that   the  quantity  has  an  inherent  positive  or  negative  attribute.   Viennot   (2004)   further   states   that   people   often  want   to  just   put   the   sign   in   front   of   the   quantity   instead   of   assigning   it   a   positive   or   negative   sign  according   to   a   chosen   coordinate   system.   An   example   of   this   can   be   seen   in   studies   (Viennot  2004;  Rebmann  &  Viennot  1994)  where  students  were  asked  to  write  the  equation  for  the  force  of  a  spring  being  contracted.  In  this  example,  students  often  “make  up  for”  the  contraction  in  the  derived  equation  ending  up  with  the  incorrect  equation  𝐹 = +𝑘𝑥  (Figure  1).    

Figure  1:  Presented  with  this  image,  students  were  asked  to  write  the  equation  for  the  force  of  the  spring  in  the  three  last  pictures.  Students  noted  that  the  force  was  directed   to  the   right   in   cases  2  and  4  and   incorrectly  put   a  +   sign   in   front  of   the  expression.  (Viennot  2004)  

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Another  example  of  the  complexity  in  signing  at  the  introductory  level  of  physics  is  in  two  body  problems   where   the   motion   is   in   different   directions,   for   example   Atwood   Machine   type  problems.  Figure  2  illustrates  the  type  of  approach  that  Brunt  (1998)  is  critical  of  because  it  fails  to  systematically  use  paired  coordinate  systems  to  establish  the  assigned  signs.  Instead  a  derived  rule  of  “bigger  force  minus  smaller  force  =  ma”  is  used.  

From   these   examples   it   is   possible   to   understand   how   students   may   not   always   have   an  appropriate  conceptualization  of  the  signs  they  are  using,  although  their  application  of  some  rule  may  lead  them  to  the  correct  answer!    

In   the   area   of   kinematics   there   have   been   specific   studies   conducted,   apart   from   the   above  mentioned,  with  the  aim  of   identifying  difficulties  students  experience  trying  to  understand  the  physical  concepts  and  how  to  connect  this  to  real  world  phenomenon.  For  example,  Trowbridge  and  McDermott   (1980;   1981)   investigated   students’   understanding   of   the   concepts   of   velocity  and   acceleration   through   interviews   with   introductory   physics   students   and   came   to   the  conclusion   that   students   often   confuse  position  with   velocity   and   velocity  with   acceleration.   It  has   further   been   reported   (Bowden   et   al.   1992)   that   a   seemingly   correct   interpretation   of  concepts  during  undergraduate  physics  courses  is  not  a  guarantee  that  students  understand  the  underlying   principles.   Bowden   et   al.   found   that   the   level   of   conceptual   understanding   often  decreased  as  the  problems  students  face  become  easier  to  solve.  This  tells  us  that  students  often  are  missing   the  conceptual  understanding  of  physics   in   introductory  courses,  which  becomes  a  large  obstacle  to  overcome  when  learning  more  advanced  physics.    

Another  challenge  students  encounter  with  introductory  physics  emerges  when  being  introduced  to  the  use  of  vectors.  For  example,  Aguirre  (1988)  discusses  students’  preconceptions  in  vector-­‐kinematics  that  often  remain  with  the  students  for  a  long  time.  The  study  investigated  students’  preconceptions  of  vectors  that  often  are  not  discussed  by  instructors  because  they  are  argued  to  be  obvious.  Aguirre  suggests  that  simply  telling  the  students  the  correct  way  of  thinking  will  not  change   their   beliefs   but   the   instructor   needs   to   be   aware   of   the   different   understandings   of  vector  conceptions  students  already  have.    

To   address   such   learning   challenges   as   described   in   the   paragraphs   above   a   physics   teacher  needs  to  have  insight  into  the  variation  of  perception  that  they  can  expect  to  find  in  their  classes.  This  study  aims  to  contribute  to  understanding  this  variation.    

 

 

 

Figure   2:   An   illustrative   Atwood   Machine   type   problem   taken   from   an   online   AP  Physics   B   lessons   (onlearningcurve   2012).   In   this   solution   no   coordinate   system   is  directly  used.  

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3 Methodology When  planning  and   conducting  a  project  of   any  kind,   the  methodology   is   an   important  part   to  consider.  This  section  will  describe  the  theory  behind  the  method  used  for  this  project  as  well  as  the  implementation  of  the  theory.    

3.1 Theory Govender’s  original   two  studies  used  a   research  approach  called  phenomenography   in  order   to  find   the  qualitatively  different  ways   in  which   students   experience   the  use  of   algebraic   signs   in  vector-­‐kinematics.   The   sorting   of   the   data   for   this   project   took   on   this   phenomenographic  perspective.  Phenomenography  is  a  research  tool  that  aims  to  describe  the  qualitatively  different  ways   individuals   experience   various   phenomena   or   aspects   in   the   world   around   them   (for  example,  see,  Trigwell  2000;  Marton  1981;  Marton  &  Booth  1997).  It  should  be  emphasized  that  phenomenography   is   not   a   research   theory,   nor   is   it   a  method   even   though   it   uses   aspects   of  both.   Instead,   Marton   and   Booth   (1997;   111)   refer   to   phenomenography   as   “a   way   of   –   an  approach   to   –   identifying,   formulating   and   tackling   certain   sorts   of   research   questions,   a  specialization  that  is  particularly  aimed  at  questions  of  relevance  to  learning  and  understanding  in  an  educational  setting”.    

Phenomenography   seeks   to   find   the   variation   in   ways   that   people   experience   a   specific  phenomenon,   and   aims   to   sort   the   experiences   into   categories   that   are   qualitatively   different  from  each  other.  This  is  done  in  order  to  find  the  limited  number  of  qualitatively  different  ways  a  phenomenon  is  experienced.    The  experiences  of  a  specific  phenomenon  are  called  categories  of  description   and   are   “the   fundamental   results   of   a   phenomenographic   investigation”   (Marton  &  Booth   1997;   122).   Further,   the   categories   of   description   can   be   listed   in   a   hierarchical  way   to  form  what   is  called   the  outcome  space  of   the  phenomenon  (Trigwell  2000).  What  characterizes  phenomenography  is  this  ranking  of  the  understandings  of  a  particular  phenomenon  (Bowden  et  al.  1992),  where  a  high  ranking,  indicates  a  better  understanding  of  the  phenomena.    

The   perspective   of   a   phenomenographic   study   is   of   the   second   order,   meaning   that   the  researcher  will   report   the  experiences  as  described  by  others   (Marton  1981).  This  approach   is  different   to   a   first-­‐order   perspective   where   the   researcher   describes   the   phenomenon   as  experienced   by   themselves.   It   is   important   to   remember   that   when   having   a   second-­‐order  perspective,   the  researcher  may  not  always  agree  with   the  experience  of   the  phenomenon,  but  the   experience   is   still   “recorded   as   a   valid   experience”   (Trigwell   2000;   6).   In   this   study,   the  second-­‐order  perspective  will  be  maintained  through  using  the  students’  experiences  as  the  base  of  analysis,  regardless  of  how  much  we  agree  with  them.    

Trigwell   (2000;  3)   gave  an  excellent   summary  of   the  phenomenographic   research  approach  as  follows:  

it  takes  a  relational  (or  non-­‐dualist)  qualitative,  second-­‐order  perspective,  that  it   aims   to   describe   the   key   aspects   of   the   variation   of   the   experience   of   a  phenomenon   rather   than   the   richness  of   individual   experiences,   and   that   it  yields   a   limited   number   of   internally   related,   hierarchical   categories   of  description  of  the  variation.    

To   investigate   how   relevant   Govender’s   original   outcome   space   is   for   current   introductory  physics   students,   a   phenomenographic   perspective   is   used   to   sort   experiences   of   the   usage   of  algebraic  signs  in  vector-­‐kinematics  from  students  in  Sweden  and  South  Africa.    

 

 

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3.2 Method The  data  that  was  used  for  this  study  was  collected  through  both  a  questionnaire  and  discussion  based   interviews.   The   questionnaire  was   given   to   students   in   the   natural   science   preparatory  programme   (basåret)   at   Uppsala   University,   Sweden,   as   well   as   to   third   year   students   in   the  physical   science   preservice   teachers’   programme   at   the   University   of   KwaZulu-­‐Natal,   South  Africa.   The   main   part   of   the   data   was   collected   through   the   questionnaires   whereas   the  interviews   focused  on  obtaining  a  deeper  understanding  of  some  of   the  answers  received   from  the  questionnaires.    

The  total  number  of  students  who  were  registered  on  the  Physics  2  course  in  basåret  in  Uppsala  was  120,  of  which  60  answered  the  questionnaire.  Unfortunately  we  have  no  data  on  how  many  students   were   actually   present   at   the   time   when   the   questionnaires   were   handed   out,   hence  there  can  be  no  statistics  showing  the  actual  response  rate  of  the  participating  Swedish  students.  However,   as   50   %   of   the   total   students   enrolled   in   the   Physics   2   course   answered   the  questionnaire,  it  can  be  seen  that  a  significant  number  of  students  from  basåret  did  take  part  in  the  study.  Among  the  South  African  students  the  response  rate  was  78  %;  a  total  of  32  students  were   on   the  Physical  Method  2   course   of  which  24   students   answered   the  questionnaire.   This  gave  us  a  total  of  84  students  completing  the  questionnaire.    

Five  Swedish   students  were   selected   to   take  part   in   semi-­‐structured   follow-­‐up   interviews.  The  five  were  chosen  from  the  28  students  that  provided  their  e-­‐mail  address  and  thereby  accepted  being   contacted   to   take   part   in   this   interview.   In   total   14   Swedish   students   were   contacted,  however  only  five  responded.  Among  the  South  African  students,  six  were  selected  to  take  part  in  an   interview.   The   selection   was   made   by   Govender   and   aimed   at   interviewing   students   from  different  races  common  in  the  KwaZulu-­‐Natal  province,  to  ensure  equity  and  representativity  of  the  student  population.  The  distribution  among  the  races  was:  one  white  student,  one  Indian  and  four  black  students.    

To   conduct   this   study,   the   method   was   divided   into   several   phases   where   the   first   phase  consisted   of   a   literature   review   of   previous   research   done   with   students’   conceptualizations  when  working  with  signs   in  physics.  The  second  phase   involved  the  creation  of  an  appropriate  ethical  agreement  to  be  used,  which  was  based  on  the  ethical  guidelines  set  up  by  the  Swedish  Research   Council   (Vetenskapsrådet   2002).   In   the   third   phase,   the   work   by   Govender   (1999;  2007)   was   considered   in   order   to   design   a   validated   questionnaire   to   give   to   the   targeted  Swedish   and   South   African   students.   The   fourth   and   final   phase   involved   conducting   semi-­‐structured  follow-­‐up  interview  discussions  (Kvale  1996)  with  purposeful  samples  (Patton  1990)  of  participating  students.    

To  be  able  to  perform  a  scientific  study  of  this  kind,  it  is  important  to  be  familiar  with  the  area  of  research   and   aware   of   any   previous   research   that   has   been   done.   Thus,   a   detailed   literature  review   was   conducted   in   the   beginning   of   the   project   to   obtain   a   full   background   for   the  particular   area   of   study.   A   thorough   background   in   issues   involved   in   performing   qualitative  research   was   also   extremely   important   in   order   to   be   able   to   design   the   questionnaire   and  interviews  in  a  suitable  way.  Before  the  questionnaire  and  interviews  could  be  carried  out  many  aspects   had   to   be   considered   to  make   sure   that   the   data   being   collected   was   appropriate   for  answering   the   research  question.   Govender   agreed   to   act   as   an   external   expert   to   validate   the  questionnaire.    

3.2.1 Ethics When   performing   scientific   research   it   is   important   to   consider   the   ethics   of   the   study.   To  maintain   the   physical   and   psychological  well   being   of   the   individuals   being   part   of   a   scientific  research   in   the   area   of   humanities   and   social   sciences,   the   Swedish  Research  Council   has   four  

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main   requirements   that   must   be   considered   for   conducting   research   of   this   kind  (Vetenskapsrådet  2002).  The  four  ethical  requirements  of  the  Swedish  Research  Council  are:  

1. The   requirement   for   information   states   that   the   researcher   has   to   inform   the  participants  of  the  aim  of  the  study.  

2. The  requirement  for  approval  means  that  the  participating  individuals  have  to  decide  for  themselves  if  they  agree  to  be  a  part  of  the  study.  

3. The  requirement  for  confidentiality  tells   the  researcher  that  he  or  she  has  to  handle  all   personal   information   from   the   participating   individuals   in   a   way   that   others  cannot  access.  

4. The  requirement  for  usage  states  that  the  information  obtained  during  the  study  will  only  be  used  for  research  purposes  and  may  not  be  used  for  non-­‐research  purposes.      

All  of   these  demands  were  met  through  the   information  given  on  the  questionnaire  and  during  the  interviews.    

3.2.2 Validity and reliability In  order  to  argue  that  the  result  of  this  study  would  be  of  scientific  value,  the  method  used  had  to  have  a  well-­‐established  credibility,  meaning  that  the  method  had  to  provide  a  valid  and  reliable  result.    

Validity  of  the  method  means  that  the  method  used  will  provide  the  result  that  is  wanted  for  this  study.  The  result  wanted  for  this  study  was  answers  from  students  explaining  their  experiences  of   the   sign   conventions   used   in   kinematic   problem   solving.   Thus,   the   method   chosen   for   this  study  had   to  provide   these  kinds  of   answers.  To  obtain  validity  of   the  method  used,  Govender  agreed  to  review  the  questionnaire,  as  well  as   the  questions  used   for   the   interviews,  during  all  the  design  stages,  thus  the  validity  of  the  method  is  argued  to  be  satisfied.      

Reliability  of  the  study  means  that  the  result  of  the  study  will  have  to  be  able  to  be  obtained  again  if   the  study   is   repeated  using   the  exact  same  method  under   the  same  conditions.   In  qualitative  research,   difficulties   can   arise   when   arguing   for   the   reliability   of   the   method.   For   example,  Merriam  (1995;  2009)  reminds  the  reader  of  the  high  improbability  that  one  will  obtain  the  exact  same   results   twice,   due   to   the   qualitative   research   being   based   on   the   experiences   of   human  beings.  Thus,  Lincoln  and  Guba   (1985)   replace   the   term  reliability  with   the   term  dependability.  The  question   to  be  asked   is   therefore   that  of   “whether   the  results  are  consistent  with   the  data  collected”  (Merriam  2009;  221).  This  means  that  from  the  data  collected,  the  result  obtained  will  have  to  make  sense.   In   this  report   I  provide  a  clear  and   full  account  of   the  research  process  so  that  the  dependability  of   the  study  can  be  assessed  and  show  how  the  research  decisions  were  made  and  implemented.      

3.2.3 Questionnaire The   main   data   that   was   collected   for   this   study   was   collected   through   a   specially   designed  questionnaire  that  was  given  to  basår  students  at  Uppsala  University,  Sweden,  and  to  students  in  the   third   year   of   the   physical   science   preservice   teachers’   programme   at   the   University   of  KwaZulu-­‐Natal,   South  Africa.  The  questionnaire   can  be   found   in  appendix  1.   Since   the   time   for  this  project  was  limited  to  ten  weeks,  this  had  to  be  considered  when  designing  the  method  for  data  collection.  With  the  use  of  a  paper  based  questionnaire  many  responses  would  be  able  to  be  collected   in   a   relatively   short   period   of   time   and   thus   this  method  was   argued   to   be   the  most  suitable   to   use.   In   order   to   have   good   data   for   the   comparison   with   Govender’s   categories,   a  special  effort  was  made  to  attract  as  many  students  as  possible  to  participate   in  the  study.  The  design  of  the  questionnaire  was  of  extreme  importance  in  order  to  maintain  a  good  quality  of  the  collected  data  and  leave  room  for  as  little  personal  evaluation  as  possible  (Robson  2002).    

To  act  as  the  foundation  for  the  study,  the  data  collected  from  the  questionnaire  should  provide  a  

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large  amount  of  different  explanations  of  how  students  use  algebraic   signs   in  vector-­‐kinematic  problem  solving..  To  limit  the  possibility  of  the  questionnaire  evoking  cognitive  overload  and/or  generating  reluctance-­‐to-­‐complete,  the  number  of  questions  had  to  be  limited.  The  questionnaire  thus   consisted  of   two  problems,   each   containing   a  number  of   questions,   to   get   the   students   to  reveal   how   they   conceptualize   the  way   they   use   algebraic   signs   in   kinematic   problem   solving.  Govender  agreed  to  act  as  an  external  expert  to  validate  the  questionnaire  and  took  part  in  all  the  design   stages.   Pilot   studies   were   done   as   a   pre-­‐test   of   the   questionnaire   (van   Teijlingen   &  Hundley  2001)  to  help  evaluate  the  adequacy  of  the  chosen  research  method,  and  where  found  to  be  necessary,  modify  the  questionnaire  outline  (see  Section  3.2.3.2).    

The   distribution   of   the   questionnaires   among   the   Swedish   students   was   done   by   the   author,  whilst  Govender  distributed  the  South  African  questionnaires.    

3.2.3.1 Design To  obtain   the  desired  data,   the   design   of   the   questionnaire  was   critical.   In   order   to   be   able   to  make   a   qualitative   comparison   with   Govender’s   five   categories,   the   problems   included   in   the  questionnaire  were  chosen  to  be  similar  to  the  problems  used  in  Govender’s  original  study.  Two  problems,  which  were  based  on  the  type  of  problems  included  in  Govender’s  study,  were  used  for  the   questionnaire.   The   problems   each   consisted   of   several   questions   exploring   how   students  apply   algebraic   signs   across   a   set   of   one-­‐dimensional   kinematics   problems   dealing   with  displacement,  velocity  and  acceleration.    

The  design  of  the  questionnaire  was  based  on  the  following:  

• time  for  completion;  • complexity  of  questions;    • possibility  to  obtain  theoretical  result.    

When   administrating   a   self-­‐completion   questionnaire   there   are   some   aspects   that   have   to   be  considered  in  order  to  obtain  the  best  result  (Robson  2002).  The  length  of  a  questionnaire  of  this  kind  is  a  critical  factor  and  should  be  kept  short.  Hence,  a  range  of  15-­‐20  minutes  for  completing  the  questionnaire  was  striven  for.  Since  there  was  no  or  little  possibility  to  clarify  the  questions  after   the  questionnaire  had  been  distributed,   the  questions  had  to  be  clearly  stated.  Because  of  this  no  problems  containing  calculations  were  included  because  this  might  scare  some  students  off.  Also,  the  questionnaire  had  to  be  designed  in  a  way  that,  theoretically,  it  would  be  possible  to  find  all  five  categories  in  the  answers  obtained.    

The   first   problem   dealt   with   a   small   ball   rolling   on   a   smooth   surface.   After   rolling   a   certain  distance  the  ball  hit  a  barrier  and  returned  to  its  original  position.  There  was  no  frictional  force  acting  on  the  ball  as  well  as  no  energy-­‐loss  in  the  collision.  Hence,  the  students  should  focus  on  the  kinematics  of   the  problem   instead  of  dealing  with   the  energy   conservation  of   the   collision.  The  students  were  asked  to  describe  the  displacement,  distance,  speed,  velocity  and  acceleration  of   the  ball   (speed  and  distance  were   included  to  provide  differential  aspects   for   the  analysis,   if  needed).  They  were  also  asked   to  describe   the  motion  of   the  ball   and  explain   if   there  was  any  difference  in  the  motion  of  the  ball  before  and  after  it  hit  the  barrier.  In  all  questions  they  were  asked  to  explain  the  meaning  of  any  algebraic  signs  they  may  have  used.  This  was  a  critical  aspect  of   the   questionnaire   since   the   study   sought   to   understand   the   ways   that   students   experience  signs,  and  not  if  the  students  were  right  or  wrong.    

The   second   problem   involved   describing   the   velocity   and   acceleration   of   a   police   car   chasing  another  car.  The  students  were  asked  to  sketch  the  velocity  and  acceleration  for  the  police  car,  using  signs  and/or  arrows,  in  five  different  parts  of  the  sequence  described,  which  involved  the  police  car  speeding  up  and  slowing  down  as  well  as  driving  at  a  constant  velocity.  The  problem  dealt  with   the  motion   in   two   opposite   directions  with   the   aim   of   investigating   if   the   students  showed  any  difference  in  their  use  of  signs  in  the  two  different  directions.  As  in  the  first  problem,  

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the  students  were  asked  to  explicitly  explain  the  meaning  of  any  signs  they  may  have  used.    

Part  of  the  aim  of  doing  all  the  pilot  studies  (see  next)  was  to  create  a  questionnaire  design  that  would   yield   the   kind   of   data   needed   to   effectively   answer   the   research   question.   In   the  main  study  the  questionnaires  carried  out  this  function  extremely  well.      

3.2.3.2 Pilot studies Several  pilot  studies  of  the  questionnaire  were  carried  out  to  obtain  as  much  feedback  about  the  design,   language,  difficulty,  and  length  as  possible.  These  studies  also  provided  feedback  on  the  validity  and  the  dependability  of  the  chosen  research  method  (see,  Peat  et  al.  2002;  123).  As  part  of  this  process,  ambiguous  questions  were  located,  questions  were  re-­‐phrased  for  clarity,  and  the  time  taken  for  completion  were  given  due  consideration.      

The  first  pilot  study  was  performed  with  two  PhD  students  in  the  group  and  tested  the  very  first  draft  of  the  questionnaire.  Several  changes  were  made  to  the  problems  used,  the  outline  and  the  design   of   the   questionnaire.   The   second   pilot   study   was   performed   with   seven   engineering  students.   The   comments   obtained   resulted   in   shortening   the   questionnaire   from   three   to   two  problems  because  it  took  the  students  a  longer  time  than  planned  to  answer  the  questions.  After  these   changes  a   final  pilot   study  was  done  with   four  engineer   students,   and   from   this,   smaller  clarifications  were  made  in  the  problem  outlines.    

3.2.3.3 Translation The  questionnaire  was  originally  written  in  English  to  make  sure  that  everyone  involved  in  the  study  would  be  able  to  follow  the  progress  on  the  design  of  the  questionnaire.  Before  distributing  it  to  the  Swedish  students,  the  questionnaire  was  translated  by  the  author.  The  translation  was  cross-­‐checked  by   three   independent  people   for   validation  before  printing.  The   translation   into  Swedish  was  argued  to  be  necessary  for  two  main  reasons;  (1)  the  participating  students’  current  physics   education   is   held   in   Swedish   and   a   questionnaire   in   Swedish   would   help   them  understand   the   questions,   and   (2)   students   that   are   familiar   with   the   physics   included   in   the  problems,   might   have   difficulties   answering   because   they   are   unfamiliar   with   the   English  language  used.    

3.2.4 Interviews Based  on  the  analysis  of  the  questionnaire  data  from  both  the  Swedish  and  South  African  student  populations,  a  group  of  students  were  selected  for  interviews  based  upon  recognizable  diversity  and  variation  in  the  questionnaire  results.  The  aim  was  to  obtain  interpretive  clarifications  that  I  had  made  of   the  descriptions   that   the  students  had  provided   in   the  questionnaire.  At   the  same  time   they  were  also   intended   to  provide  deeper   insight   into   the  descriptions   that   the   students  provided.  .  Swedish  students  that  were  willing  to  take  part  in  an  interview  provided  their  e-­‐mail  address   with   their   questionnaire   and   were   contacted   with   an   enquiry   to   take   part   in   an  interview.    

The   Swedish   interviewees  were   purposefully   selected   (Patton  1990)   using   one   or  more   of   the  following  criteria:  

• showed  interesting  experiences  in  at  least  one  question;  • explained  their  reasoning  carefully  in  a  majority  of  the  questions;  • shared  the  same  experiences  throughout  the  questionnaire;  • shared  the  same  experiences  throughout  the  questionnaire  but  deviated  from  this  in  one  

or  more  questions.    

Since   the   South   African   students   came   from  many   different   schools   throughout   the   KwaZulu-­‐Natal  province  before  they  attended  the  university,  the  participating  students  were  purposefully  selected  to  capture  a  variation  of  these  students  and  thus  of  answers  to  the  questionnaire.    

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The  type  of  interview  used  for  the  study  was  semi-­‐structured  (Kvale  1996).  Thus,  questions  were  predetermined,  however   the  order  of   the  questions  could  be  changed  and  additional  questions  added   to   follow   up   on   the   interviewee’s   answers.   During   the   interviews,   the   students   were  presented  with  his  or  her  questionnaire  allowing  the  interviewer  to  address  specific  answers  or  wording  for  clarification.    

3.2.4.1 Design The   interviews   were   designed   to   clarify   and   deepen   my   understanding   of   the   students’  experiences   of   the   use   of   algebraic   signs   in   kinematic   problem   solving.   The   interviews   were  designed   to   be   short,   not   involving   questions   that   would   give   too   much   room   for   long  descriptions.  The  interviews  were  kept  in  the  order  of  15-­‐20  minutes,  which  put  high  demands  on  the  interviewer  not  to  let  the  interview  lose  focus.  The  interviews  were  kept  this  short  also  to  attract  students  to  participate.    

The  questions  used  for  the  interviews  were  chosen  to  give  the  students  a  chance  to  freely  explain  their  experiences  about  algebraic  signs  used  in  the  problems,  however  keeping  the  topic  specific.  The   predetermined   questions   were   of   three   different   types:   (1)   “what   sign   would   you   use   to  describe   displacement/speed/velocity/acceleration?”,   (2)   “how   does   the   sign   for  velocity/acceleration   change   during   the   car   chase?”,   (3)   “what   does   the   sign   for  velocity/acceleration  mean  to  you  in  everyday  life/physics  etc.?”.    

The  first  of  the  three  types  of  questions  were  mostly  directed  to  the  first  problem  included  in  the  questionnaire  while   the   second   type  was  directed   to   the   second  problem.  The   third   type  were  questions   that   would   provide   an   explanation   of   the   students’   overall   experience   of   algebraic  signs  used  in  different  contexts.  Further  questions  asked  during  the  interviews  depended  on  the  situation   and   varied   between   the   different   interviews   but   aimed   to   clarify   interesting  experiences  from  the  questionnaire  or  to  follow  up  on  answers  given  during  the  interview.    

During   the   interviews  a  commonly  used  sequence  of  questions   for   interviews  was  mainly  used  (Robson   2002),   opening  with   an   introduction   to   the   study,  which   included   the   purpose   of   the  study  and  explained  the  ethics  involved.  All  interviews  were  recorded,  after  permission  from  the  participants,  which  allowed  the  interviewer  to  focus  entirely  on  engaging  with  the  participants.    

3.2.4.2 Pilot interview Before  the  first  interview  the  interview  protocol  was  tested  on  an  independent  person  to  identify  any  ambiguity  in  the  questions  and  for  the  interviewer  to  feel  more  secure  when  doing  the  later  interviews.  During   the  pilot   interview,   the   same  questions   and   the   same   setup  were  used   that  were  going  to  be  used  for  the  student  interviews.    

The  pilot  interview  provided  further  understanding  of  how  to  conduct  interviews  and  supported  the  planning  and  conducting  of  the  student  interviews.  Difficulties  with  the  interviews  that  were  highlighted   during   the   pilot   interview   included   knowing  when   it  was   appropriate   to  move   on  with  the  next  question  as  well  as  how  to  keep  a  neutral  tone  signaling  that  there  was  no  prejudice  towards  the  students’  understandings.    

3.2.4.3 Transcriptions All  Swedish  and  South  African  interviews  were  transcribed  verbatim  by  the  author.  Transcribing  the   interviews  very  carefully  ensured   that  no   important   information  was  omitted.  The  extracts  from   the   interviews  used  within   this   report  have  been   translated  when  necessary  but  without  the  meaning  of  the  question  or  answer  being  changed.    

3.2.5 Analysis The   analysis   of   the   data   was   done   in   line   with   Govender’s   (1999;   2007)   research   where   the  obtained   answers   from   both   questionnaires   and   interviews   were   sorted   in   the   five   already  

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existing   qualitatively   different   categories   (see   appendix   2).   However,   possibility   for   the  appearance  of  any  new  category  was  taken  into  account  during  the  process.  Following  Govender,  a  phenomenographic  perspective  was  used  when  sorting   the  data   for   the  analysis   (see  Section  3.1).   It   is   important   to   emphasize   that   throughout   the   analysis   no   references  were  made   to   a  specific  student,  but  rather  to  a  specific  experience  of  the  use  of  algebraic  signs.    

The   analysis   process   of   the   data   was   performed   in   many   different   stages   and   culminated   in  sorting  the  questionnaire  answers  and  interview  transcripts  into  boxes  representing  each  of  the  five  original  categories.  Each  questionnaire  was  given  a  specific  number  instead  of  the  students’  name  to  uphold  the  confidentiality  and  the  interview  transcripts  were  given  the  same  number  as  the  corresponding  questionnaire  for  that  particular  student.  This  was  done  for  both  the  Swedish  and   South   African   data.   After   being   numbered,   the   individual   answers   on   the   questionnaires  were   cut   apart   leaving   them   each   on   a   single   piece   of   paper.   The   same   thing   was   done   for  interesting   excerpts   from   the   interview   transcripts.   Throughout   the   analysis   process   this  numbering,   linking  questionnaires  with   interviews,  was   crucial   in   order   to  make   it   possible   to  find  the  answers  if  one  of  the  participating  students  asked  to  withdraw  his  or  her  answers.    

To   begin   the   analysis,   the   first   set   of   data,   being   the   Swedish   questionnaire   answers,   were  roughly  divided   into  groups  showing  similar  experiences  of  algebraic  signs.  These  groups  were  further   analyzed   being   split   into   new   groups   or   conjoined   with   others,   a   process   that   was  conducted   several   times.   Next,   the   answers   from   each   group  were   sorted   into   physical   boxes  representing   the   five   qualitatively   different   categories.   This   sorting   was   remade   through   an  iterative  process  until  the  categories  had  been  saturated,  meaning  that  no  categories  were  placed  in   another   box   when   resorted.   The   sorting   and   resorting   of   the   questionnaire   answers   were  made  with  a  few  days  in  between.    

Once  the  first  set  of  data  had  been  analyzed,  all  other  data,  meaning  Swedish  and  South  African  interview   transcripts   and   South   African   questionnaire   answers,   was   only   analyzed   through  sorting   and   resorting   directly   into   the   physical   boxes.   This   reduction   of   analysis   stages   was  possible   because   through   the   first   analysis   process   it   had   become   easier   to   recognize   the  different  categories  and  thus  there  was  no  need  to  make  the  first  rough  sorting.  The  Swedish  and  South  African  data  were  sorted  in  separate  boxes  to  be  able  to  make  a  comparison  between  the  experiences  of  the  two  student  populations.    

In   all   stages   of   the   data   sorting   and   categorization,   the   same   phenomenographic   research  perspective  was  used.    

 

4 Results In   the   following   sections   the   results   of   the   study   will   be   presented.   The   analysis   from   the  questionnaires   and   interviews   will   be   compared   to   the   outcome   space   generated   from  Govender’s  (1999;  2007)  study  (appendix  2).    

First,  the  results  of  the  questionnaire  and  interviews  will  be  given  as  a  summary  of  the  findings,  followed  by   an   evaluation   of   the   results.   Later,   example   excerpts   from   the   different   categories  will   be  presented   in  order   to  better  describe   the   categories.  Here,   no  differences  will   be  made  between  the  Swedish  and  South  African  students.  The  excerpts  named  “Question”  and  “Answer”  originate   from   a   questionnaire,   while   the   excerpts   coded   “Interviewer”   and   “Student”   shows  examples  of  answers  from  an  interview.    

 

 

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4.1 Summary of findings From   the   analysis   of   the   data   obtained   from   the   questionnaires   and   interviews,   I   was   able   to  identify   four   of   the   five   categories   proposed   by   Govender   (2007)   to   describe   the   variation   of  ways  students  experience  the  use  of  algebraic  signs  in  vector-­‐kinematics  in  both  sets  of  data.  No  new  categories  were   found.  The  obtained  outcome   space   for  my   study   is  presented   in  Table  1  below.  The  names  of  the  categories  are  the  same  as  those  in  Govender’s  original  study.  

Table  1:  The  obtained  outcome  space    

A:   Algebraic  signs  are  not  applied  in  vector-­‐kinematics  

C:   Algebraic  signs  are  applied  as  changing  magnitude    

D:   Algebraic  signs  are  applied  as  both  magnitude  and  direction  

E:   Algebraic  signs  are  applied  as  directions  

The   relative   frequency   of   each   categorization   is   shown   in   Figure   3   using   a   scale   of   1-­‐4.   It   is  important   to   point   out   that   a   categorization   of   a   piece   of   description   is   not   to   be   taken   to   be  synonymous  with   a   single  person;   a  person  may  have  provided  descriptions   that   fit   into  more  than  one  category.  The  most  commonly  used  description  in  both  the  Swedish  and  South  African  data  sets  fell  into  Category  E  followed  by  Category  C.  The  two  categories  that  were  the  next  most  common  were  Categories  A  and  D.  Here,  Category  A  was  slightly  more  common  in  the  Swedish  data  set,  while  Category  D  was  more  common  in  the  South  African  data.  Govender’s  Category  B  (Algebraic   signs   are   applied   as  magnitude   only)   was   not   found   from   the   analysis   of   either   the  Swedish  or  the  South  African  data  sets,  hence  this  category  is  not  included  in  the  figure.    

Some  answers  were  not  analysable   for   this   study  since   they  did  not  contain   information  about  the  use  of   algebraic   signs,   or  were   completely  blank.   Particularly   in   the   second  problem  many  answers   could   not   be   analysed.   In   this   problem   students   were   asked   to   sketch   the  velocity/acceleration   of   the   police   car   during   the   chase   with   the   use   of   signs   and/or   arrows.  Here,   many   students   only   drew   arrows   without   any   signs   and   thus   these   answers   were   not  analysable  for  this  study.    

0   1   2   3   4  

A  

C  

D  

E  

Relative  frequency  in  data  sets  

Category  

South  African  data  set  

Swedish  data  set  

Figure  3:  The  relative  frequency  of  the  categories  occurring  in  the  Swedish  and  South  African  data  sets.  The  scaling  used  is  1-­‐4  where  4  is  the  most  common  and  1  the  least  common.  Govender’s  original  Category  B  (Algebraic  signs  are  applied  as  magnitude  only)  is  not  included  because  it  was  not  found  among  the  analysed  answers.    

 

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Among   the   South   African   students   almost   all   answers   were   analysed,   while   many   Swedish  students  failed  to  answer  all  questions.  The  question  that  was  left  blank  most  often  was  question  2.5.    

4.2 Evaluation The   results   obtained   showed   that   not   all   of   Govender’s   original   categories,  which   characterize  the  ways  that  students  conceptualize  how  algebraic  signs  should  be  used  in  kinematics  problem  solving,   could   be   found   among   the   student   populations   involved   in   this   study;   Swedish  introductory   physics   students   and   South   African   physical   science   preservice   teacher   students.  The  obtained  outcome  space  of  the  qualitatively  different  ways  students  apply  algebraic  signs  in  kinematics  problem  solving  can  be  found  in  Table  1.    

The   research   question   asked   how   relevant   Govender’s   categories   were,   that   is   if   they   were  generalizable   to   another   context.   Being   able   to   find   only   four   of   Govender’s   five   original  categories,   I   can  conclude   that  Govender’s   results  were  not  generalizable  as  an  entire  outcome  space   for   the   qualitatively   different   ways   students   experience   algebraic   signs   in   vector-­‐kinematics.    

When   analysing   the   data   obtained,   it   was   found   that   the   experiences   among   the   Swedish   and  South  African  students  was  the  same,  which  implies  students  in  different  contexts  share  the  same  experiences  and/or  challenges  in  the  area  of  vector-­‐kinematics.    

Since  this  study  has  been  able  to  identify  four  qualitatively  different  categories  of  description  of  how   students   conceptualize   the   use   of   algebraic   signs   in   vector-­‐kinematics,   it   is   of   interest   to  investigate  how  teaching  and  learning  has  developed  during  the  15  years  that  has  passed  since  Govender’s   first   study.   Although   the   reason   for   not   finding   one   of   the   original   categories  (Algebraic  signs  are  applied  as  magnitude  only)   is  something  that  we  presently  cannot  comment  on,  we  would  like  to  think  that  the  development  of  teachers'  and  physics  instructors’  teaching  has  been  the  foundation  for  this  change  in  conceptualization.    

4.3 Description of categories Next   a   presentation   of   the   findings   from   this   study   will   be   made   showing   the   qualitatively  different  ways  students  in  the  study  experience  the  use  of  signs  in  vector-­‐kinematics.  As  it  is  not  possible   to   display   all   the   different   conceptions   belonging   in   the   qualitatively   different  categories,  and  thus  the  excerpts  shown  below  will  only  highlight  some  interesting  aspects  found.  The   different   quotes   will   show   the   variation   of   experiences   in   the   same   category   to   highlight  various  interpretations.    

A: Algebraic signs are not applied in vector-kinematics The  first  category  proposed  by  Govender  states   that  students  experience  algebraic  signs  as  not  applied   at   all   in   vector-­‐kinematics   to   describe   the   concepts   of   displacement,   velocity,   and  acceleration.   This   category   shows   that   students   make   no   connection   to   the   representation   of  vectors-­‐concepts   in   terms  of   signs.  Below  are   some   interesting  explanations   for   the  absence  of  algebraic  signs  obtained  from  the  questionnaires  and  interviews.    

Question:     Is  direction  important  to  specify  the  motion  of  the  ball?  Explain.  Answer:     Yes,  the  direction  is  important  to  specify  because  it  tells  you  where  the  rolling  ball  

comes  from.  We  need  to  tackle  how  to  give  its  direction  of  motion,  because  positive  or  negative  will  not  work  since  it  could  be  moving  at  any  angle.  

This  student  is  unable  to  connect  signs  as  a  description  of  directions  of  a  vector.  The  connection  to  direction  is  made,  however  the  student  feels  that  plus  and  minus  is  not  sufficient  to   indicate  

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the  direction  of  motion  of  the  ball,  since  there  can  be  many  different  angles.  Here,  students  do  not  connect  plus  and  minus  to  a  coordinate  system.    

Another  example  of   the  unwillingness   to  use  signs   to  describe  vector-­‐concepts   is   shown   in   the  excerpt   below.   The   student   does   not   show   any   need   or   motivation   to   use   signs   to   describe  vector-­‐concepts  and  cannot  remember  having  seen  this  used  before.    

Question:     Explain   the  meaning  of   any  algebraic   sign   (+  or   -­‐)   that   you  used   [to   illustrate   the  velocity  of  the  police  car].    

Answer:     You   cannot   use   arrows   or   signs   to   describe   the   velocity   or   acceleration,   only  numbers.   I   have   at   least   never   encountered   anything   other   than   numbers   to  describe  this  within  physics.    

This  clearly  shows  us  that  the  student  has  not  made  the  connection  between  algebraic  signs  and  direction.  Unfortunately  this  excerpt  comes  from  a  student  who  was  not  interviewed  and  thus  a  clarification  of  this  experience  could  not  be  obtained.    

Further,   students   often   show  willingness   to   express   the   direction   of   a   vector   (or   the  motion),  however  not  particularly  in  terms  of  plus  and  minus.    

Interviewer:    When  you  say  that  𝑎 = 5  left,  you  don’t  say  that  𝑎 = −5?  Student:     No.    Interviewer:    Why  not?  Student:     It   depends   on   me.   As   long   as   I   in   the   beginning   would   specify   that   left   means  

something  is  going  to  the  left.    

Question:   Describe  the  speed  and  velocity  of  the  ball  before  and  after  the  collision.  Explain  the  meaning  of  any  algebraic  sign  (+  or  -­‐)  that  you  used.    

Answer:     As  before,  a  motion  to  the  right  feels  positive  and  to  the  left  negative.  I  now  realize  that  I  think  that  +  and  –  seems  a  bit  unnecessary.  Why  don’t  you  just  say  a  motion  to  the  right  or  left?    

This  experience  shows  that  students  do  not  link  algebraic  signs  with  directions  and  feel  that  they  are  redundant   to  describe  the  motion  of  an  object.  The  students  see   it  as  sufficient   to  state   the  direction  in  already  defined  terms  such  as  left  or  right,  instead  of  using  signs  that  to  the  students  have  no  explicit  meaning  in  the  problem  solving  process.    

C: Algebraic signs are applied as changing magnitude A   common   experience   of   the   use   of   algebraic   signs   in   vector-­‐kinematics   was   that   signs   are  applied  as  denoting  the  change  of  magnitude.  One  student  explained  this  explicitly  as  follows:  

“I   experience   plus   as   something   that   is   getting   bigger,   and   minus   as   something   that   is   getting  smaller”.    

This  category  was  mostly  used  to  describe  the  change  in  magnitude  for  velocity  and  acceleration.  The  different  meanings  of  the  signs  applied  for  change  in  velocity  and  signs  applied  for  change  in  acceleration  is  interesting  to  point  out.    

Question:     Explain  the  meaning  of  any  algebraic  signs  (+  or  -­‐)   that  you  used  [to   illustrate  the  velocity  of  the  police  car].    

Answer:     For  increasing  motion  the  positive  sign  was  used,  regardless  of  the  direction,  while  a  negative  sign  was  used  for  decreasing  velocity.    

Question:     Explain  the  meaning  of  any  algebraic  signs  (+  or  -­‐)   that  you  used  [to   illustrate  the  acceleration  of  the  police  car].    

Answer:   (+)  Increase  in  acceleration,  (-­‐)  Decrease  in  acceleration.    

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The   above   excerpts   show   that   students   use   plus   and   minus   as   changing   magnitude   for   both  velocity  and  acceleration.  An  interesting  similarity  can  be  found  between  the  use  of  signs  for  the  two  conceptions.  Note  that  the  signs  denote  the  change  in  magnitude,  i.e.  the  change  in  velocity  and   the   change   in   acceleration.  However,  while   it   is   easy   to   picture   a   change   in   velocity   as   an  object  is  speeding  up  or  slowing  down,  a  change  in  acceleration  is  harder  to  picture.  The  meaning  of  this  will  be  discussed  in  Section  5.    

Another   interpretation   of   signs   linked   to   a   change   in  magnitude   is   the   next   excerpt.   Here   the  student  was   asked   to   illustrate   the   acceleration  of   the  police   car   in   the   second  problem  of   the  questionnaire:    

Question:     Explain  the  meaning  of  any  algebraic  signs  (+  or  -­‐)  that  you  used.    Answer:     +𝑎  means  the  car  is  accelerating.  −𝑎  means  the  car  is  decelerating.  

Interviewer:     [What  sign  would  you  use]  for  acceleration?  Student:     When  it’s  acceleration,  it’s  speeding  up,  I  would  say  positive  and  when  it’s  slowing  

down  I  would  say  negative.  

From  the   first  excerpt  we  see  that  students  of   this  experience   link  a  negative  acceleration  with  deceleration,  meaning  a  decrease  in  velocity.  The  interpretation  of  this  is  that  to  students  of  this  experience,  a  negative  acceleration  will  always  imply  a  deceleration,  regardless  of  the  direction  of   the   motion   of   the   object.   This   interpretation   is   closely   linked   to   the   second   excerpt   above  where  a  negative  acceleration  was  used   to  describe  an  object   slowing  down.  From   this  we   see  that  for  students,  a  negative  acceleration  can  be  experienced  as  a  decrease  in  velocity,  the  object  is  slowing  down,  regardless  of  the  motion  of  the  object.    

A   final   interesting   excerpt   for   this   category   shows   the   explicit  meaning   of   the   use   of   algebraic  signs  as  denoting  change  in  magnitude.  This  student  is  consistent  in  his  reasoning  and  does  not  differentiate  between  the  use  of  algebraic  signs  in  everyday  life  and  physics.    

Interviewer:   What  does  plus  and  minus  mean,  in  general?  Student:   I  would  say  change  in  magnitude.    Interviewer:   And  in  physics,  what  does  plus  and  minus  mean  to  you?  Student:   The   direct   experience   is   that   it   means   bigger   or   smaller.   It   is   not   linked   to  

directions.    

D: Algebraic signs are applied as both magnitude and direction The   category   stating   that   algebraic   signs   are   applied   as   both   magnitude   and   direction   is   an  inappropriate  experience  of  the  use  of  algebraic  signs  in  vector-­‐kinematics.  This  category  shows  that   students   experience   algebraic   signs   in   different   ways   for   different   concepts;   often   this  difference  is  displayed  between  velocity  and  acceleration.  In  other  words,  algebraic  signs  are  not  used  consistently  in  vector-­‐kinematics.    

In  the  excerpts  below,  it  is  shown  how  students  interpret  algebraic  signs  as  both  magnitude  and  direction.    

Question:     Is   there  any  difference  between  the  velocity  and  acceleration  arrows  or  signs  that  you  drew  in  the  above  questions?  If  so,  explain.    

Answer:     Yes,   in   velocity   the   signs   only   specify   the   direction   of   motion,   however   in  acceleration  it  means  speeding  or  slowing.    

Question:   Is   there  any  difference  between  the  velocity  and  acceleration  arrows  or  signs   that  you  drew  in  the  above  questions?  If  so,  explain.    

Answer:   Yes,   when   it   comes   to   velocity   +   and   –   only   show   direction.   When   it   comes   to  acceleration  they  only  show  the  acceleration’s  increase  or  decrease  and  doesn’t  take  direction  into  consideration.  Why  it  turned  out  this  way  I  don’t  know!    

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These   excerpts   exemplify   how   students   have   little   understanding   of   the  meaning  of   the  use   of  signs   in   vector-­‐kinematics.   These   students  have   the   right   interpretation  of   the  use  of   signs   for  describing   the  velocity  of   an  object   (plus  and  minus  are  applied  as  directions),  however   in   the  second   excerpt   the   student   shows   confusion   about   how   signs   can   mean   different   things   for  velocity   and   acceleration.   The   interpretation   tells   us   that   students   are   unaware,   or   have   little  understanding,  of  the  sign  conventions  used  in  vector-­‐kinematics.    

The   excerpt   below   is   another   example,   obtained   from   one   of   the   interviews,   illustrating   how  students  describe  the  use  of  algebraic  signs  as  both  magnitude  and  direction.      

Interviewer:    What  does  the  velocity  sign  mean  to  you  in  general?    Student:     The  velocity  signs  mean  directions.    Interviewer:    And  acceleration?    Student:     For  acceleration  they  mean  whether  speeding  up  or  slowing  down.  For  speeding  up  

I  use  plus,  for  slowing  down  minus.    

E: Algebraic signs are applied as directions The  only  appropriate  experience  among  the  five  categories  described  by  Govender  was  the  last  category  stating  that  signs  are  applied  as  directions  only.  As  in  the  other  categories,  students  may  have  different   interpretations  of   signs,   thus   in   this   case  directions  have  different  meanings   for  different  experiences.  For  example,  “directions”  may  mean  direction  of  the  motion  or  direction  of  the  vector  quantity.    

A  common  experience  of  signs  applied  as  directions  is  that  signs  are  dependent  on  the  direction  of  the  movement,  rather  than  the  direction  of  the  vector  quantity.    

Question:     Describe  the  acceleration  of  the  ball  before  and  after  the  turn.  Explain  the  meaning  of  any  algebraic  signs  (+  or  -­‐)  that  you  used.    

Answer:     The   acceleration   will   be   positive   during   forward   motion   and   negative   during  reverse  motion.  

We   note   that   when   assigning   the   signs   the   student   only   takes   the   direction   of   motion   into  consideration  and  does  not  link  the  signs  to  the  direction  of  the  vector.  Unfortunately  the  student  does   not   explain  why   the   signs   are   applied   this  way.   The   following   excerpts,   obtained   from   a  different  student,  further  explain  this.  

Question:     Explain   the  meaning  of  any  algebraic   signs   (+  or   -­‐)   that  you  used   [to  describe   the  acceleration  of  the  police  car].  

Answer:     Only  +  is  used  since  there  is  only  one  direction,  i.e.  there  is  no  change  of  direction  to  the  opposite  direction  because  the  car  slowing  down  doesn’t  change  the  sign.    

During  the  following  interview  with  the  same  student,  clarification  was  sought.    

Interviewer:   How  does  the  sign  of  the  acceleration  of  the  police  car  change?  Student:   It  is  positive.  Because  as  long  as  it  is  following  the  Volvo  in  one  direction  I  take  it  as  

positive  which  means  it  is  going  to  the  right.    Interviewer:   So  when  it  accelerates  and  slows  down  you  would  give  the  same  sign?  Student:   Yes  

This,   although   an   incorrect   use   of   signs   as   directions,   primarily   shows   that   the   student   is  consistent   in   his   interpretation.   The   student   allocates   signs   to   describe   the   acceleration  depending  on  the  direction  of  the  car.  Thus,  the  experience  is  that  signs  are  applied  as  directions,  however  not  the  direction  of  the  vector  quantity  as  would  be  the  correct  experience.    

Often,  in  both  questionnaires  and  interviews,  students  expressed  that  plus  and  minus  applied  as  directions   were   just   something   they   had   been   taught   and   they   did   not   further   think   of   the  meaning  of  this.    

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Question:     Is  there  any  difference  to  the  motion  of  the  ball  before  and  after  the  turn?  Explain  the  meaning  of  any  algebraic  signs  (+  or  -­‐)  that  you  used.  

Answer:     We   usually   determine  motions   to   the   right,   the   original  motion,   as   positive,   thus  with  a  +  sign.  When  the  ball  has  turned,  it  travels  in  negative  (-­‐)  direction  compared  to  its  original  motion.    

Question:     Is  there  any  difference  to  the  motion  of  the  ball  before  and  after  the  turn?  Explain  the  meaning  of  any  algebraic  signs  (+  or  -­‐)  that  you  used.  

Answer:     I  have  learned  that  right  is  indicated  with  a  +  and  left  with  a  -­‐.  

We   see   that   students   might   not   think   about   the   meaning   of   the   chosen   sign   convention,   but  accept  it  as  something  that  is  defined  to  be  this  way.  They  think  that  “right”  is  connected  to  +  and  “left”   is   connected   to   –   in   all   cases  without   understanding  why.   That   students   accept   the   sign  conventions  presented  to  them  by  their  teachers  can  be  seen  in  the  excerpt  below.  .    

Interviewer:    Why  was  the  motion  of  the  ball  positive  to  start  with?  Student:     I  guess  that  it  is  just  something  I  assume  or  presuppose.    Interviewer:    And  then  it  becomes  negative,  why?  Student:     Well,   after   it   turned   it   can’t   continue  have   the   same   sign   as  before.   I   think   that   it  

should  be  negative.    

To   the   student,   different   directions   should   be   described   with   different   signs.   However,   no  understanding  of  the  meaning  or  the  background  to  this  is  seen.    

 

5 Discussion The  analysis  of  the  data  obtained  from  students  in  the  basår  programme  at  Uppsala  University,  Sweden,   and   physical   science   preservice   teacher   students   at   the   University   of   KwaZulu-­‐Natal,  South  Africa,  has  revealed  that  four  of  the  five  qualitatively  different  categories  of  how  students  experiences   the   use   of   algebraic   signs   in   vector-­‐kinematics,   reported   on   by   Govender   (1999;  2007),  still  are   found  today.  The  category   that  was  not   found  among  the  participating  students  was  Category  B  (Algebraic  signs  are  applied  as  magnitude  only).    

Through   analysis   of   the   data   I   have   been   able   to   sort   the   descriptions   into   categories   of   the  variation  of  ways  students  in  Sweden  and  South  Africa  apply  algebraic  signs  in  vector-­‐kinematics.  This   analysis   led   to   identifying   four   of   the   five   categories   from  Govender’s   original   study.   The  obtained  outcome  space  of  these  qualitatively  different  categories  of  experiences  can  be  found  in  Table  1.  I  have  not  only  been  able  to  show  the  existence  of  these  four  categories  across  these  two  contexts   15   years   later,   but   can   also   say   that   the   Swedish   and   South   African   data   sets   were  similar.  Hence   the  categories   found   is   claimed   to  be  consistent   in  different   contexts.  What  also  should  be  emphasized  is  that  Govender’s  study  was  carried  out  nearly  15  years  ago,  showing  that  students’  experience  of  signs  has  not  changed  significantly  over  a  long  time  period.      

Four   of   the   five   original   categories  were   identified.   This  means   that   the   full   generalizability   of  Govender’s   categories   was   not   established   by  my   study.   This   does   not  mean   that   the  missing  category  no  longer  exist,  a  larger  study  could  perhaps  show  that  this  category  still  exist.  

Even   though   I  was   not   been   able   to   find   Category   B   (Algebraic   signs  are  applied  as  magnitude  only)   among   the  data  obtained   in   this   study,   from  a  phenomenographic  perspective  we  cannot  with   certainty   exclude   this   category   from   the   outcome   space   of   how   students   experience  algebraic   signs   in   vector-­‐kinematics.   Since   phenomenography   seeks   the   variation   of   ways  through  a  qualitative  rather  than  a  quantitative  approach,  this  category,  since  it  has  been  found  in  an  earlier  study,  cannot  be  eliminated  completely.  It  has  not  been  found  in  this  study,  but  there  is  a  possibility  that  it  could  be  found  in  future  studies.    

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A   question   that   now   naturally   arises   is   why   we   were   not   able   to   identify   all   categories.   The  absent  category  was  found  15  years  ago  in  South  Africa,  but  could  not  be  found  in  either  of  the  data  sets.  Since  we  have  no  research  showing  the  development  of  the  physics  education  in  South  Africa  we  cannot  draw  any  conclusions  about  the  elimination  of  the  category.    

Using   the   results   of   this   study,   we   will   be   able   to   inform   physics   teachers   about   students’  difficulties   when   using   signs   in   vector-­‐kinematics,   which   will   provide   teachers   with   valuable  pedagogical  tools  to  improve  physics  education  that  will  better  suit  the  diversity  of  the  student  populations   that   can   be   found   at   introductory   physics   levels   today.   Since   it   has   been   shown,  through   this   and   other   research   (see   for   example,   Govender   1999;   Hayes   &  Wittmann   2010;  Aguirre   1988;  McDermott   1984),   that   students   experience   difficulties  with   vectors   and   vector  notations  it  is  important  that  teachers  become  aware  of  students’  struggles  in  order  to  help  them  overcome  their  challenges.  If  teachers  are  not  familiar  with  the  common  problems  that  students  may  have,  it  will  be  difficult  to  design  a  physics  education  that  meets  their  students’  need.    

It   is   not   hard   to   recognise   that   if   teachers   are  not  made   aware   of   the   struggles   students   have,  there  will  not  be  a  change  in  the  teaching  and  learning.  One  of  the  goals  with  this  study  was  to  suggest   different   pedagogical   tools   for   teachers,   based   on   the   results   of   the   study,   to   improve  physics  education.  However,  unfortunately  there  has  not  been  enough  time  to  further  investigate  this.  Nevertheless,   I  will  below  discuss  common  challenges   I  have   identified  and  suggestions  of  what  might  be  the  causes  of  these  interpretations  of  signs.    

From  doing  this  study  I  have  come  to  better  understand  the  challenges  students  encounter  when  being   introduced   to   vectors.   I   could   recognise  myself   in  many   of   the   answers  which  made  me  remember   the   difficulties   I   experienced   when   learning   how   to   use   vectors.   To   be   able   to  recognise  my  own  thinking  in  the  data  made  me  feel  a  strong  link  to  the  study  and  made  me  even  more   eager   to   obtain   a   result   that   would   inform   teachers.   I   have   realised   that   when   you   use  vector   notation   appropriately   after   several   years   of   physics   training,   it   might   be   difficult   to  recognise   difficulties   that   students   are   encountering   or   to   understand   what   causes   these  difficulties.  This  is  why  it  is  important  that  teachers  are  informed  about  the  results  of  this  study  to  help  their  students  overcome  their  difficulties.    

5.1 Identified challenges Below  I  will  discuss  some  of  the  educational  challenges  that  have  been  highlighted  through  this  study  and  will  briefly  discuss  suggestions  of  how  to  help  students  overcome  these  challenges.    

Students are not consistent with their use of algebraic signs From  Category  D  which  states  that  algebraic  signs  are  used  as  both  magnitude  and  direction,   it  can   be   seen   that   students   are   inconsistent   in   their   use   of   signs.   On   one   hand   signs   indicate   a  direction  and  on   the  other  hand,  plus  and  minus  are  applied  as  signifying  changing  magnitude.  When  analysing  individual  questionnaires,  it  was  found  that  the  same  student  described  the  use  of  plus  and  minus  differently   in  different  problems,  which  might  be  an   indication  that  students  use  different  sign  conventions  in  different  problem  contexts.    

To   help   students   overcome   this   incorrect   use   of   algebraic   signs,   I   believe   it   is   important   that  teachers  are  clearer  about  the  reasons  for  using  signs  but  also  that  students  get  to  meet  vectors  in  different  ways  to  help  them  understand  the  sign  conventions  used.    

Students apply an incorrect use of signs in the same way for both velocity and acceleration As   seen   in   the   description   of   Category   C   (Algebraic   signs   are   applied   as   changing  magnitude),  students  often  indicate  that  signs  are  used  to  describe  an  increase  or  decrease  in  velocity.  From  a  physics  point  of  view,  this  is  an  inappropriate  use  of  signs  in  vector-­‐kinematics  which  shows  that  

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students   are   not   familiar   with   the   common   sign   conventions   used.   Another   common  interpretation  of  signs  in  this  category  was  that  plus  and  minus  indicates  an  increase  or  decrease  in  acceleration.  This  is  also  an  inappropriate  use  of  signs  since  it  shows  that  for  the  students  it  is  not  clear  how  to  interpret  a  change  in  acceleration  in  terms  of  signs.    

Students’   incorrect  use  of  signs  in  the  same  way  (as  changing  magnitude)  for  both  velocity  and  acceleration,  might  originate  from  an  incorrect  interpretation  of  signs  applied  for  acceleration.  If  students  have  learned  that  the  sign  for  acceleration  indicates  an  increase  or  decrease  in  velocity,  they  might   think   that   signs   are   used   in   the   same  way   to   describe   the   velocity.   For   example,   if  students  have   learned   that  plus   is  used   for  acceleration  when  a   car   is   speeding  up,   they  might  think   that   the   same   sign   will   also   be   used   for   velocity   in   the   same   situation.   Hence,   to   these  students   there   is   no   experienced   difference   in   the   application   of   signs   between   velocity   and  acceleration.    

The  same  approach  might  be  the  explanation  for  the  interpretation  that  signs  for  acceleration  are  applied  as  direction  of  the  motion.  As  we  could  see  in  Category  E,  many  students  assigned  plus  and  minus   signs   for   acceleration   indicating   in  what  direction   the  object  was  moving,   instead  of  the  correct  interpretation  that  signs  should  indicate  the  direction  of  the  acceleration  vector.  This  might  be  explained  in  a  similar  way  as  in  the  paragraph  above.    

If  students  have  learned  that  signs  are  used  for  velocity  to  indicate  in  what  direction  the  object  is  moving,   they  might   think   that   signs  are  used   in   the   same  way  also   for  acceleration.  Hence,   the  sign  for  acceleration  indicates  the  direction  of  the  motion.  For  example,  if  an  object  is  moving  to  the  right,  a  plus  sign  is  used  to  describe  its  velocity,  similarly  a  plus  sign  is  used  to  describe  the  acceleration  of  the  object,  regardless  if  the  object  is  speeding  up  or  slowing  down.      

Students need to experience vectors in more than one way Many   students   described   the   use   of   plus   and  minus   in   a  way   that   tells   us   that   often   teachers  define   the   “correct”   sign   conventions   without   telling   students   this   is   an   arbitrary   choice.  Examples   of   this   could   be   found   in   Category   E   (Algebraic   signs   are   applied   as   directions).   By  telling  students  that  “right  is  positive”  and  “left  is  negative”  they  only  come  to  see  vectors  as  used  in  one  way  hence,   the  correct  understanding  of  algebraic  signs   in  vector-­‐kinematics  will  not  be  clear  to  them  even  though  they  might  be  using  signs  as  directions.  To  overcome  this  problem,  I  think  it  is  important  that  students  are  tested  more  on  their  conceptual  understanding,  something  that  is  also  proposed  by  McDermott  (1984).    

“Direction” means different things in different situations The  most   common  category   that   could  be   found  among   the   student   responses  was  Category  E  stating  that  signs  are  applied  as  directions.  This  is  proposed  by  Govender  as  the  appropriate  use  of  signs  in  vector-­‐kinematics.  This  category  was  expected  to  be  the  most  common  when  starting  the  study,  which  would  mean  that  students  do  make  the  connection  between  algebraic  signs  and  directions   for   vector-­‐concepts,   just   as   we   hoped   they   would.   We   need   to   discuss   the   many  different  ways  students  use  signs   for  directions,  not  only   the  direction  of   the  vector  arrow,  but  the  direction  of  motion  or  direction  of  change  of  magnitude.    

As  described   above,   signs   applied   as  directions   can  be   interpreted   in  different  ways.  A   correct  interpretation   is   that   the   sign   for   velocity   is   linked   to   the   direction   of   motion.   However   this  interpretation   is  not  appropriate   to  be  used   for   the  concept  of  acceleration.  Despite   this,  many  students  applied  the  same  sign  for  acceleration  that  was  used  to  describe  the  direction  of  motion.  Another  inappropriate  use  of  signs  applied  as  directions  is  when  signs  are  applied  as  a  direction  of  change  of  magnitude,  as  discussed  previously.  Students  might,  for  example,  use  plus  and  minus  to  denote  the  direction  of  the  change  in  velocity  meaning  that  if  the  velocity  decreases  the  change  is  directed  in  the  negative  direction  and  the  velocity  becomes  negative.    

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From  the  above  examples   it   can  be  seen   that   “direction”  can  mean  different   things   to  different  students,  especially  in  different  contexts.  Even  though  students  are  aware  that  signs  are  used  as  directions,  they  might  not  realize  that  directions  mean  the  direction  of  the  vector.  Thus  teachers  need  to  be  more  explicit   in  explaining  that  signs  are  used  as  the  direction  of  the  vector  and  not  the  motion  or  change  in  magnitude.    

Students do not understand the reason for using signs Category  A  (Algebraic  signs  are  not  applied  in  vector-­‐kinematics),  describes  students’  experience  where  they  have  not  all  grasped  the  sign  convention  used  and  do  not  understand  the  signs  used  in   vector-­‐kinematic   problem   solving.   Teachers   need   to   make   clear   to   students   that   plus   and  minus  is  a  tool  for  describing  vector-­‐quantities  in  a  coordinate  system.  Hence,  it  is  important  that  teachers  help  the  students  to  differentiate  between  the  directions  in  everyday  life  (right  and  left)  and  the  directions  in  physics  (plus  and  minus).    

 

6 Recommendations It   is   proposed   that   this   research   is   developed   in   future   studies   in   order   to   verify   the   results  obtained.   This   should   include   a   close   investigation   to   see   if   Category   B   (Algebraic   signs   are  applied   as   magnitude   only),   which   could   not   be   found   in   this   study,   could   be   found   using   a  different  data  set.  A  continuation  of   this  project  should  also   involve   further  studies  of  how  this  result   should   be   implemented   in   the   teaching   and   learning   of   introductory   physics.   It   is  important  to  find  out  how  teachers  can  take  this  result  and  use  it  to  develop  their  teaching.  Also,  it  is  of  interest  to  find  out  more  about  the  reason  for  the  students’  conceptual  difficulties,  which  can   be   found   through   learning  more   about   how   teachers   explain   and   demonstrate   the   use   of  signs  in  vector-­‐kinematics  in  the  classroom.    

It  is  important  that  teachers  understand  the  results  of  this  study  in  order  for  them  to  be  aware  of  students’   difficulties   and   experiences   of   algebraic   signs   in   vector-­‐kinematics   and   also   how  common   these   are   for   them   to   be   able   to   improve   the   teaching   and   learning   of   introductory  physics.  A   central   issue   for   teachers   to   grasp   is   the  necessity   of   being   clearer   about   the   actual  meaning   of   algebraic   signs   as   the   direction   of   the   vector   and   not   the   direction   of   the   quantity  itself.    

 

7 Conclusions In   order   to   find   the   number   of   qualitatively   different   ways   students   of   introductory   physics  experience  the  use  of  algebraic  signs  in  vector-­‐kinematics  to  be  able  to  examine  the  relevance  of  Govender’s  results,  I  designed  a  special  questionnaire  where  students  were  asked  to  explain  their  use  of   signs   in   one-­‐dimensional   kinematics  problems.  The  questionnaire  was   completed  by  60  Swedish  basår  students  and  24  South  African  physical  science  preservice  teacher  students.  Based  on  the  answers  from  the  questionnaire,  students  were  purposefully  selected  to  take  part  in  semi-­‐structured  interviews,  where  I  sought  clarifications  to  some  experienced  interpretations  of  signs.  In  total  five  Swedish  students  and  six  South  African  students  took  part  in  the  interviews.    

All   questionnaire   answers   and   interesting   excerpts   from   the   transcribed   interviews   were  analysed  using  sorting  based  on  a  phenomenography  perspective,  where  the  researcher  tries  to  find  the  variation  of  ways  people  experience  a  phenomenon.  Through  the  analysis,  I  was  able  to  identify  four  of  the  five  original  categories  proposed  by  Govender  (1999;  2007).  The  categories  can  be  found  in  Table  1.    

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The   study   has   primarily   shown   that   the   outcome   space   of   how   students   experience   algebraic  signs  in  vector-­‐kinematics  found  by  Govender,  has  only  slightly  changed  since  the  first  study  was  performed  15  years  ago.  The  category  that  could  not  be  found  in  my  study  stated  that  students  experience   that   algebraic   signs   are   applied   as  magnitude   only.   The   reason   for   this   need   to   be  investigated   in   future   studies.   Further,   I   have   also   been   able   to   show   that   in   this   study   no  differences  were  seen  between  the  experiences  of  algebraic  signs  among  the  students  in  Sweden  and  South  Africa  and  the  obtained  outcome  space  could  be  used  in  different  contexts.    

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References Aguirre,  J.  M.  (1988).  Student  preconceptions  about  vector  kinematics.  The  Physics  Teacher,  26,  

212-­‐216.    

Bowden,  J.,  Dall’Alba,  G.,  Martin,  E.,  Laurillard,  D.,  Marton,  F.,  Masters,  G.,  Ramsden,  P.,  Stephanou,  A.,  &  Walsh,  E.  (1992).  Displacement,  velocity,  and  frames  of  reference:  Phenomenographic  studies  of  students’  understanding  and  some  implications  for  teaching  and  assessment.  American  Journal  of  Physics,  60,  262-­‐269.    

Brunt,  G.  (1998).  Questions  of  sign.  Physics  Education,  33(4),  242-­‐249.    

Goldberg,  F.  M.,  &  Anderson,  J.  H.  (1989).  Student  difficulties  with  graphical  representations  of  negative  values  of  velocity.  The  Physics  Teacher,  27,  254-­‐260.  

Govender,  N.  (1999).  A  phenomenographic  study  of  physics  students  experience  of  sign  conventions  in  mechanics.  (unpublished  PhD  thesis,  University  of  the  Western  Cape,  South  Africa)  

Govender,  N.  (2007).  Physics  student  teachers'  mix  of  understandings  of  algebraic  sign  convention  in  vector-­‐kinematics:  A  phenomenographic  perspective.  African  Journal  of  Research  in  Science,  Mathematics  and  Technology  Education,  11(1),  61-­‐73.  doi:  http://dx.doi.org/10.1080/10288457.2007.10740612  

Hayes,  K.,  &  Wittmann,  M.  C.  (2010).  The  role  of  sign  in  students’  modeling  of  scalar  equations.  The  Physics  Teacher,  48(4),  246-­‐249.  

Henderson,  C.,  Dancy,  M.,  &  Niewiadomska-­‐Bugaj,  N.  (2012).  Use  of  research-­‐based  instructional  strategies  in  introductory  physics:  Where  do  faculty  leave  the  innovation-­‐decision  process?  Physical  Review  Special  Topics  -­‐  Physics  Education  Research,  8(2),  020104.  

Kvale,  S.  (1996).  Interviews:  an  introduction  to  qualitative  research  interviewing.  Thousand  Oaks,  CA:  Sage.    

Lincoln,  Y.  S.,  &  Guba,  E.  G.  (1985).  Naturalistic  inquiry.  Beverly  Hills,  CA:  Sage.    

Marton,  F.  (1981).  Phenomenography  -­‐  Describing  conceptions  of  the  world  around  us.  Instructional  Science,  10(2),  177-­‐200.    

Marton,  F.,  &  Booth,  S.  (1997).  Learning  and  awareness.  New  Jersey:  Lawrence  Erlbaum  Associates.    

McDermott,  L.  C.  (1984).  Research  on  conceptual  understanding  in  mechanics.  Physics  Today  37(7),  24-­‐32  

Merriam,  S.  B.  (1995).  What  can  you  tell  from  an  N  of  1?:  Issues  of  validity  and  reliability  in  qualitative  research.  PAACE  Journal  of  Lifelong  Learning,  4,  51-­‐60.  

Merriam,  S.  B.  (2009).  Qualitative  research:  A  guide  to  design  and  implementation.  San  Francisco,  CA:  Jossey-­‐Bass.    

onlearningcurve  2012,  AP  Physics  B  Lessons  with  Ms.  Twu:  Forces  9:  Multi-­‐Object  Problems  Part  3:  Atwood  Machine,  online  video,  viewed  24  February  2014,    https://www.youtube.com/watch?v=ceWAX-pNqC4.  

Patton,  M.  Q.  (1990).  Qualitative  evaluation  and  research  methods.  Newbury  Park,  CA:  Sage.  

Peat,  J.,  Mellis,  C.,  Williams,  K.,  &  Xuan  W.  (2002).  Health  Science  Research:  A  Handbook  of  Quantitative  Methods.  London:  Sage.  

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Rebmann,  G.,  &  Viennot,  L.  (1994).  Teaching  algebraic  coding:  Stakes,  difficulties  and  suggestions.  American  Journal  of  Physics,  62(8),  723-­‐727.  

Robson,  C.  (2002).  Real  world  research:  a  resource  for  social  scientists  and  practitioner-­‐researchers,  2  ed.  Oxford:  Blackwell.    

Testa,  I.,  Monroy,  G.,  &  Sassi,  E.  (2002).  Students'  reading  images  in  kinematics:  the  case  of  real-­‐time  graphs.  International  Journal  of  Science  Education,  24(3),  235-­‐256.  

Trigwell,  K.  (2000).  Phenomenography:  Variation  and  Discernment.  In  C.  Rust  (ed)  Improving  Student  Learning,  Proceedings  of  the  1999  7th  International  Symposium,  Oxford  Centre  for  Staff  and  Learning  Development:  Oxford,  75-­‐85.  

Trowbridge,  D.  E.,  &  McDermott,  L.  C.  (1980).  Investigation  of  student  understanding  of  the  concept  of  velocity  in  one  dimension.  American  Journal  of  Physics,  48,  1020–1028.    

Trowbridge,  D.  E.,  &  McDermott,  L.  C.  (1981).  Investigation  of  student  understanding  of  the  concept  of  acceleration  in  one  dimension.  American  Journal  of  Physics,  49,  242–253.    

van  Teijlingen,  E.  R.,  &  Hundley,  V.  (2001).  The  importance  of  pilot  studies.  Social  Research  Update,  35,  1-­‐4.    

Vetenskapsrådet  (2002).  Forskningsetiska  principer  inom  humanistisk-­‐samhällsvetenskaplig  forskning.  Stockholm:  Elanders  Gotab.    

Viennot,  L.  (2004).  Reasoning  in  physics:  the  part  of  common  sense.  New  York:  Kluwer  Academic  Publishers.  

 

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       Appendix 1 The  questionnaire  that  was  distributed  among  the  participating  students.    

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 Problem  1:  The  motion  of  a  rolling  ball    A  small  ball  rolls  along  a  smooth  surface  (ignore  friction).  When  the  ball  has  rolled  2m,  it  reverses  when  it  hits  a  barrier  (no  energy  is  lost  during  the  collision)  and  it  rolls  back  to  its  original  position.  For  the  questions  below,  please  explain  your  reasoning  carefully.      

1.1:  Is  there  any  difference  to  the  motion  of  the  ball  before  and  after  the  turn?  Explain  the  meaning  of  any  algebraic  signs    (+  or  -­‐)  that  you  used.              

1.2:  Is  direction  important  to  specify  the  motion  of  the  ball?  Explain.    

           

1.3:  Describe  the  distance  and  displacement  of  the  ball  before  and  after  the  turn.  Explain  the  meaning  of  any  algebraic  signs  (+  or  -­‐)  that  you  used.                    

1.4:  Describe  the  speed  and  velocity  of  the  ball  before  and  after  the  turn.  Explain  the  meaning  of  any  algebraic  signs  (+  or  -­‐)  that  you  used.                    

1.5:  Describe  the  acceleration  of  the  ball  before  and  after  the  turn.  Explain  the  meaning  of  any  algebraic  signs  (+  or  -­‐)  that  you  used.        

 

Before:   After:  

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 Problem  2:  Velocity  and  acceleration  of  a  car  chase  Imagine  the  following  sequence:      (1)  A  police  car  is  standing  by  the  side  of  the  road  at  the  intersection  between  Dag  Hammarskjölds  väg  and  Kungsängsleden  when  she  sees  a  Volvo  travelling  at  a  constant  speed  through  a  red  light.    (2)  The  police  car  immediately  starts  chasing  the  Volvo,  along  a  straight  part  of  the  road,  accelerating  from  rest  until  reaching  a  maximum  chasing  speed.  (3)  The  officer  holds  this  speed  until  she  is  alongside  to  the  Volvo.    (4)  She  turns  on  the  blue  light  signalling  to  the  Volvo  to  pull  over.  The  driver  of  the  Volvo  starts  to  slow  down,  the  police  car  also  slows  down,  staying  alongside  the  Volvo.  (5)  Both  cars  finally  stop  by  the  side  of  the  road.      2.1:  For  the  different  parts  of  the  sequence  (1)-­‐(5)  above,  sketch  the  velocity  of  the  police  car  using  arrows,  and  signs  if  appropriate.               (1)   (2)   (3)   (4)   (5)    

2.2:  Explain  the  meaning  of  any  algebraic  signs  (+  or  -­‐)  that  you  may  have  used.    

2.3:  For  the  different  parts  of  the  sequence  (1)-­‐(5)  above,  sketch  the  acceleration  of  the  police  car  using  arrows,  and  signs  if  appropriate.               (1)   (2)   (3)   (4)   (5)    

2.4:  Explain  the  meaning  of  any  algebraic  signs  (+  or  -­‐)  that  you  may  have  used.    

 

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 2.5  Supposed  the  police  car  turns  around  and  follows  the  exact  same  sequence  in  the  other  direction.      

2.5.1:  For  the  different  parts  of  the  sequence  (1)-­‐(5),  sketch  the  velocity  of  the  police  car  using  arrows,  and  signs  if  appropriate.               (1)   (2)   (3)   (4)   (5)    

2.5.2:  Explain  the  meaning  of  any  algebraic  signs  (+  or  -­‐)  that  you  may  have  used.    

2.5.3:  For  the  different  parts  of  the  sequence  (1)-­‐(5),  sketch  the  acceleration  of  the  police  car  using  arrows,  and  signs  if  appropriate.               (1)   (2)   (3)   (4)   (5)    

2.5.4:  Explain  the  meaning  of  any  algebraic  signs  (+  or  -­‐)  that  you  may  have  used.    

2.6  Are  there  any  differences  between  the  arrows  and/or  signs  that  you  used  to  describe  the  velocity  and  acceleration  respectively  in  the  above  questions?  If  so,  please  explain.    

   

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Appendix 2 Govender’s  (2007)  article  describing  the  study  conducted  and  displaying  the  result  including  the  outcome  space  of  the  qualitatively  different  ways  students  experience  the  use  of  signs  in  vector  kinematics.    

This  article  is  reproduced  as  part  of  this  project  report  with  the  permission  of  Fred  Lubben,  Chief  Editor  and  publishing  coordinator  of  the  African  Journal  of  Research  in  Mathematics,  Science  and  Technology  Education.      

 

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Physics student teachers' mix of understandings of algebraic sign convention in vector-kinematics: A phenomenographic

perspective

Nadaraj Govender School of SMTE, Edgewood Campus, University of KwaZulu-Natal

[email protected] Abstract Physics pre-service student teachers' mix of understandings of positive (+) and negative (–) algebraic sign convention in vector-kinematics was explored using three familiar demonstration contexts in physics. The students were interviewed about their understandings of algebraic sign convention as applied to high school Physical Science and Mathematics and, specifically, to fundamental concepts in vector-kinematics, namely, displacement, velocity and acceleration. The interviews were analysed from a phenomenographic perspective. Five different conceptions and hierarchical relations of the different ways student teachers' understand algebraic sign convention were identified. The data highlights the need for physics educators to take cognisance of both student teachers' limited conceptual understandings of vector-kinematics and its associated algebraic sign convention in first-year university physics. Introduction One of the important aspects of mathematical 'inscriptions' – numbers, graphs, signs, symbols etc. (Roth, Tobin & Shaw, 1997, p. 1075) are signs and sign convention, which serve as a tool or technique for expressing physical quantities and solving of physics problems. Sign convention is an arbitrary choice of mathematical representation used as a technique in problem solving, for example, algebraic signs (+) and (–) is used to denote direction of motion of object. By assigning the "+" and "–"signs to numbers one refers to what has been called directed numbers (Fischbein, 1994). In vector-kinematics this involves symbolic representations of algebraic operations and subsequent manipulation and interpretation of those symbols within the context of problem solving in one, two and three dimensions. However, research has shown that students in tertiary physics are often confused by the application of algebraic signs in vector-kinematics (Rebmann & Viennot, 1994; Romer, 1998). In this regard, Rebmann and Viennot (1994) argue that since algebraic language is one of the main tools used in physics, it is educationally beneficial to analyse students' comprehension of and skill in algebraic procedures including their notion of algebraic signs. Since physics teachers play an important role in the development of their students' understanding of fundamental concepts, it is therefore critical to develop insight into the different ways in which student teachers' as prospective physics teachers understand sign conventions in physics. A literature review of work in this area of physics has shown that qualitative studies in algebraic sign convention in vector-kinematics are rare. This study attempts to contribute to filling this gap by exploring the different ways that student teachers' conceptualise algebraic sign convention in vector-kinematics. Since the study was about identifying the range of perceptions of sign conventions in vector-kinematics, phenomenography as a research perspective was deemed the most appropriate method to analyse students' understandings. Phenomenography was developed by Ference Marton (1981) to discover the "qualitatively different ways in which people experience, conceptualise, perceive, and understand various aspects of, and phenomena in, the world around them." In this study, the phenomenon is positive (+) and negative (–) algebraic sign convention as applied to vector-kinematics concepts of displacement, velocity and acceleration.

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Tertiary pre-service physics student teachers were interviewed about their understandings of algebraic sign convention as applied to everyday Physical Science and Mathematics contexts, and to fundamental concepts in vector-kinematics, namely, displacement, velocity, and acceleration, using familiar demonstration contexts in physics. The interviews were then analysed from a phenomenographic research perspective. Literature review The role of symbols in scientific communication Positive (+) and negative (–) algebraic sign convention performs an essential role in physics, as it provides us with a succinct and efficient method for use in vector-kinematics. Reif (1995) elaborates on the use of symbols, of which sign convention is just one aspect, by pointing out that:

Formal modes of conceptions, using precisely defined symbols and explicit rules for their manipulation, are very well suited to facilitate long and accurate inference chains. Such formal modes of conceptions, exploiting mathematics and logic, are thus widely used in physics (p. 23).

Rebmann and Viennot (1994) investigated university student teachers' mastery of algebraic sign use in physics and added that correct usage and conventions are far from obvious to students. Brunt (1998, p. 242) stated that, positive and negative algebraic signs are frequently misapplied. He added that some authors of textbooks might be unaware of some of the basic rules of algebraic sign convention and, for clarification of readers, he discussed some of the rules for sign convention in physics. Romer (1998) also emphasises the latter point and says, referring to the confusion in sign convention that, "there are so many places one could go wrong that it sometimes seems that one has at best a 50-50 chance of getting it right" (Romer, 1998, p. 849). Algebraic sign convention in kinematics – a complex problem Students find great difficulty in relating the motion of an object to its variables and algebraic formalism. In an investigation with physics students in mechanics to determine correspondence between an observed motion and the algebraic formalism, Lawson and McDermott (1987, p. 811) found that students make little connection between the algebraic symbols of the formula and the features of the demonstration – unlike a physicist, who almost 'sees' the application of formula to real world situations. Newburgh (1996) also noted that first-year physics students find it difficult to understand the role of mathematics in making inferences about the real world. He suggested that this is partly because they have not developed much physical intuition. Such intuition comes from experience and that real, imaginary and complex numbers appearing as solutions in mechanics problems can pose difficulties in interpreting such solutions. There are many different symbols and notations to denote vectors, and students often get confused in using them in physics. Boldface signs, for example, +, – and = signs are used in vector equations in some textbooks to emphasis the distinction between vector and scalar operations with ordinary numbers. The importance of vector-scalar symbolism is highlighted by Arons (1997) who feels that if students do not distinguish between scalar and vector quantities in the notation, they will most likely not make the distinction in their thinking either. Allie and Buffler (1998) noted that, in South Africa, vectors are not discussed in the Mathematics school syllabus and are dealt with only in the Physical Science syllabus. Since most of the high school physics problems are dealt in one-dimensional physical situations, where vectors are labelled with positive and negative signs, "high-school students end up perceiving no practical difference between scalar and vector algebra" (Allie & Buffler, 1998, p. 618).

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Students experience confusion surrounding algebraic sign convention especially in the transition between one- and two-dimensional motion, with the notion of scalars and vectors coming into play. Scalars have a sign and a magnitude whereas vectors have magnitude, which is inherently positive, and has direction (Warren, 1979). The components of a vector in a rectangular one- two- or three-dimensional Cartesian coordinate system can have positive or negative signs. In a special case, the one-dimension Cartesian system component is taken to have direction that can have a negative or positive algebraic sign. From a mathematical perspective, a directed scalar can be considered as a one-dimensional vector. This notion is also accepted in the one-dimensional treatment of vectors in physics but the existence of both positive and negative values of a quantity in physics can sometimes be wrongly regarded as a vector. This is possibly one reason why some scalars, such as potential energy and temperature, which can have negative values, are often mistakenly thought to be vectors (Warren, 1979). Students also experience difficulties with the concepts of displacement, velocity and acceleration, and in interpreting algebraic signs for kinematical graphs of motion. Trowbridge and McDermott's (1980) study showed that there was confusion about the vector nature of velocity. The concept of acceleration, and its associated sign convention in vector-kinematics, poses a more difficult aspect for students. Warren (1979, p. 3) adds that difficulties caused by the incorrect use of the negative sign to acceleration, implying that the speed is decreasing, arise in cases where a body undergoes periodic motion, or thrown upwards, or in head-on collision of objects. Students, who are taught that rectilinear acceleration is a rate of increase of speed, and use negative rectilinear acceleration or 'deceleration' (not a formal physics term) for a rate of decrease, find it very difficult to accept, in the case of uniform circular motion, that a body is accelerating when its direction changes at a constant speed. Students' application of algebraic sign convention is further compounded by poor conceptual knowledge in mechanics. Halloun and Hestenes (1985) noted that the most common, and critical, problem for students was a failure to discriminate between various kinematical quantities. Research indicates that a significant number of students do not master the basic kinematical ideas in the first years of introductory physics (McDermott, 1984; Trowbridge & McDermott, 1980, 1981). Aquirre (1988) also suggests that teachers of introductory physics need to give explicit consideration to the study of vectors. The Mechanics Baseline Test by Hestenes and Wells (1992) reported that the lowest scores were generated by questions requiring an understanding of vector properties. Hestenes and Wells (1992) adds that while students face an immediate hurdle with the concept of force, inabilities to reason correctly about vector quantities, present a second, and less-studied, hurdle. First-year university and final year high school physics student teachers' understanding of fundamental concepts in vector-kinematics has been explored using the phenomenographic research perspective (Bowden et al., 1992; Dall'alba et al., 1993; Govender, 1999). From the analysis of the interviews from these studies, a hierarchical set of conceptions of the different ways students understand the concepts in mechanics have been developed and the relationship between different levels of understanding have been identified. More recently, Erdman (2004) citing students' difficulties with vectors devised outdoor activities to force students to apply the concept of vector in terms of length and direction. The above literature review suggests that the understanding of algebraic sign convention and concepts in vector-kinematics can be a complex issue. In physics, a sign convention is a choice of the signs (plus or minus) of a set of quantities, in a case where the choice of sign is arbitrary – meaning that the same physical system can be correctly described using different choices for the signs, as long as one set of definitions is used consistently. Thus central to a thorough understanding of sign conventions in vector-kinematics is the comprehension of the notion of arbitriness or free choice of assigning signs. Although there are several articles on vectors and kinematics over the past ten years, a recent literature search (2007) has shown that no new

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studies were published on sign convention in vector-kinematics. This may possibly be due to a notion by physicists that there is little or no difficulty experienced by first-year students in vector-kinematics or an inadequate understanding of their students' school physics background. Thus this paper contributes to a better understanding at the school-university interface by providing a deeper analysis of student teachers' qualitative understandings of algebraic sign convention that can contribute to physics education. Methodology Detailed knowledge of the ways in which students understand the central phenomena and concepts within a domain prior to study, is believed to be critical for developing their understanding of the central phenomena, and, hence, of the mastery of the domain (Bowden et al., 1992). This study thus uses the phenomenographic research perspective developed by Ference Marton (1981) to discover the "qualitatively different ways in which people experience, conceptualise, perceive and understand various aspects of, and phenomena in, the world around them". According to Marton and Booth (1997), phenomenography uncovers not the 'inner world' of the students but how the students see their relation to the world. In this study students seek to recognise and structure relationships, in the mix of conventions and definitions in kinematics, linked to the real world phenomena with which they were presented. Thus phenomenography is an appropriate methodology to identify the distinctive different ways in which we organise our relations to the world. In all phenomenographic studies, it has been found that each phenomenon, concept or principle can be understood in a limited number of qualitatively different ways. In this study, it is also assumed that a limited number of ways of understanding positive (+) and negative (–) algebraic sign convention as applied to the vector-kinematical concepts in physics can be found. This study formed part of a longitudinal qualitative case study. The study involved interviewing 19 first-year college physics pre-service student teachers in order to determine their conceptualisations of positive (+) and negative (–) algebraic sign convention used in vector-kinematics. The pre-service student teachers are prospective physics teachers pre-selected for a four-year teacher-training course. An interview protocol was used to facilitate the elicitation and collection of student teachers' understandings of algebraic sign convention in vector-kinematics. Student interviews were carried out on an individual basis and were based on three demonstrations using common material and simple apparatuses that students are likely to have encountered in everyday life. The first was 'a ball rolled on a smooth surface', the second was 'a ball thrown up and returning to its original position' and the third 'a ball was rolled up an inclined plane and returned to its original position'. In all these situations the students were required to discuss the concepts of displacement, velocity and acceleration and the application of positive (+) and negative (–) algebraic sign convention to these concepts in the context of understanding vector-kinematics in mechanics. The transcripts that originated from different individual interviews made up the data to be analysed. The data was then analysed phenomenographically. Initially, the process involved looking for comprehensive structures of use and explanation of algebraic signs in all the data and, then identifying distinct ways of conceptualising positive (+) and negative (–) algebraic sign convention in vector-kinematics. Conceptualisations and not students were being sampled. There was no attempt to fit the data into predetermined categories. The categories of conceptions of algebraic sign that were drawn from the interview data were obtained from a lengthy process of iteration. To ensure reliability of data, the process of iteration was repeated, twice, with a month gap. To ensure validity of data, the 'cut-out' slips of interview data was re-worked together with a physicist with experience in phenomenography, ensuring that the categories of conceptions were consistent to the ones obtained by the researcher. The final result is expressed as an outcome space that is defined as "the pool of meaning, a complex of categories of description comprising distinct groupings of aspects of the phenomenon and the relationships between them" (Marton & Booth, 1997, p. 125).

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Results of study Table 1 provides a general description of categories of conceptions with examples to elucidate how student teachers assigned algebraic signs to vector-kinematical concepts of displacement, velocity and acceleration. Table 2 provides the outcome space ─ the qualitatively distinct and different ways of understanding algebraic sign convention in vector-kinematics. The conceptions captured are all possible ways of understanding algebraic sign conventions in vector-kinematics; some are conceptually incorrect in terms of accepted understanding by the scientific community. Table 1: Categories of conception for positive (+) and negative (–) algebraic signs applied to displacement, velocity and acceleration

A.1 Algebraic signs are not applied to vector-kinematical concepts of displacement, velocity and acceleration. In this case definitions of concepts suffice.

B.1 Algebraic signs are used as positive and negative integer numbers represented on a number line as magnitude (in Mathematics). B.2 Algebraic signs are used to denote magnitude of speed (v) as 'how fast or how slow' e.g. v = -5 means slow or v = +5 object going fast.

C.1 Algebraic signs are used for velocity (v) and acceleration (a) to denote changing magnitude, e.g. v = +5 means the object is going faster or v = -5 means the object is slowing down.

D.1 Algebraic signs for acceleration (a) applied as direction and magnitude, e.g. a = -5 means that the object moves left and is slowing down.

E.1 Algebraic sign for direction of gravitational acceleration is negative only but direction of acceleration for straight-line motion can have both positive and negative signs. E.2 Algebraic signs applied as directions for displacement using the number-line concept with zero as the origin, e.g. left of the origin is (-) and to the right is (+). E.3 Algebraic signs applied as directions to vector-kinematical concepts without reference to zero as an origin, e.g. (+) is motion to the right and (-) motion to the left.

Table 2 provides the outcome space (the qualitatively different ways of understanding) of algebraic signs in one-dimensional vector-kinematics. Table 2: Outcome space for algebraic signs in one-dimensional vector-kinematics

Conception Outcome space for algebraic sign convention in vector-kinematics A Algebraic signs are not applied in vector-kinematics B Algebraic signs are applied as magnitude only C Algebraic signs are applied as changing magnitude D Algebraic signs are applied as both magnitude and direction E Algebraic signs are applied as directions

The hierarchical arrangement in Table 2 of the different ways of understanding algebraic signs in one-dimensional vector-kinematics provides a basis for analysing conceptions about how learning of algebraic sign convention occurs at first-year physics level. The categories of conceptions allow for the differentiation among ways of conceptualising a particular phenomenon, is this case, algebraic sign convention. Category A reflects a narrow understanding of algebraic signs in vector-kinematics whereas Categories B-D reflects mixed understandings and Category E reflects the correct scientific conception. In this study the outcome space (see Table 2) for the phenomenon of positive (+) and negative (–) algebraic signs for one-dimensional kinematical motion, provides five qualitatively different ways in which students assigned algebraic signs to vector-kinematical phenomena. This constitutes the main outcome of the research. The next section discusses in detail examples supporting these categories of conceptions.

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Examples and analysis of interviews supporting the categories of conceptions The discussion to follow examines in detail the different ways students understand algebraic sign conventions in vector-kinematics as evident in Table 1. Each category of conception is supported by sample evidence obtained from student interviews. The main purpose of matching interview extracts with categories of conceptions was to provide strong evidence linking fieldwork, that is, student teachers' conceptions from interviews of the phenomenon of algebraic sign convention, with the invention of the outcome space (Table 2). The outcome space is the second-order conceptions derived from the phenomenographic iteration process of the phenomenon of algebraic sign convention as interpreted by the researcher. A.1. Algebraic signs are not applied to vector-kinematical concepts of displacement and velocity. The student applies the definition of displacement and since the answer obtained is correct, the student does not use algebraic signs.

Interviewer: What algebraic sign would you use for displacement? Student: No sign. Because it is 5 m from starting point. It does not matter whether

it's up or down. When the object returns, the displacement is zero. Interviewer: Why zero? Student: Because from the definition, if the object returns to its starting position,

displacement is zero. The concept of displacement is possibly treated synonymously with the concept of distance. Since distance is always positive in magnitude and includes just the length and not direction, displacement is also assumed by the student to be positive and therefore cannot have a negative sign although for one-dimensional motion we do assign a negative sign for direction.

Interviewer: What algebraic signs would you use for speed and velocity? Student: No signs also. Speed is just a number and velocity is speed plus direction

that can be a compass like North. To this student, the concepts of speed, displacement, and distance are seen as inherently positive, as magnitude with a positive quantity while velocity is understood correctly as having both magnitude and direction hence a vector. The difficulty the student faces is translating the cardinal compass directions into signs, for example, representing North as (+). The student also indicates that the direction of acceleration is not represented by a sign as evident in the following extract.

Interviewer: What does the sign for acceleration mean? Student: We don't have to use sign convention for acceleration. You don't

necessarily have to use it for acceleration because it's not an actual vector thing for direction or something.

In this category, there is little evidence to show that students understand the significance of symbolic representations in physics. Thus, a shift towards a deeper conceptual understanding of algebraic signs to be interpreted as directions in one-dimensional vector-kinematical motion is necessary. B.1 Algebraic signs used as positive and negative integer numbers on a number-line as representing magnitude of vector–kinematical concepts The notion of negative numbers has confused mathematicians for centuries (Fischbein, 1994). The understanding of negative numbers now has a theoretical mathematical basis and is presented with different understandings at different levels of education. The everyday understandings of (+) and (–) signs may start from very early childhood but in South Africa,

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the formal conceptual development of positive integers (+1, +2 etc.) start in primary schools in Grade 3 (9-10-year-olds) and negative integers (–1, –2 etc.) in Grade 7 (12-13-year-olds) where a number-line extending in both right and left directions with positive and negative integers are taught in Mathematics. The number-line is taught as a mathematical representation of magnitude of numbers and is reinforced from Grades 3-9 (9-15-year-olds), a period of approximately six years.

Interviewer: Where did you use + and – signs? Student: Mainly in Mathematics, the numbers where the other side of the number

line is less than zero. In the number-line system, the numbers and their signs are examined from a theoretical perspective as defined in Mathematics. Conventionally, the number +10 is drawn on the right of the number line implying a positive number. The –10 implies a negative number and smaller in magnitude. Integers have magnitude only and no direction (scalars) but in one-dimensional vectors, the signs of the numbers (–1 or + 1) are assigned directions and the number, a positive magnitude. In calculating displacement in one-dimensional vector-kinematical problems, students commonly use scalar subtraction rather than vector addition to determine the displacement of an object. While we are familiar that in one dimension, scalar subtraction has the same result as vector addition, students lack the conceptual understanding that calculations with scalar quantities use the operations of ordinary arithmetic, but vector addition requires a different set of operations and, in particular, that subtraction of vectors means to add vectors considering both their magnitude and direction. B.2 Algebraic signs are used to denote magnitude for speed as magnitude of 'how fast or how slow'. For speed, the student interprets algebraic signs as magnitude (how fast or slow) rather than as a direction. Thus a value of +10 m.s-1 simply means a positive magnitude (related to size or quantity) that the object is moving with a certain speed. The (–) sign is interpreted incorrectly as 'slow' whereas one should compare different values of speed. For example, –5 m.s-1 is incorrectly interpreted as the object moving slowly whereas one should compare the speed of 10 m.s-1 to 5 m.s-1 .

Interviewer: Would you use signs for speed and velocity? Student: It might confuse them if I give them for speed – using direction they

would think that speed also has direction. They would be confused because if I am using sign convention for speed, I will be using plus (+) for ... how fast or how slow minus (–)

C.1 Algebraic signs for velocity and acceleration are used as changing magnitude.

Interviewer: What does the sign for acceleration mean? Student: We can use it as negative or deceleration. If the answer came out to be

negative then that is deceleration, it's slower in that sense. This category implies that 'going faster' is assigned a positive (+) sign and 'going slower' a negative (–) sign. The sign for acceleration is understood by the student as obtained from the changing magnitude of the velocity. Thus the sign of acceleration is often incorrectly interpreted as follows; a positive sign if the object is going faster or negative sign if the object is going slower. This seems to be a common interpretation by students and this notion is sometimes applied to both the concepts of velocity and acceleration; hence the vector-kinematical concepts are not clearly differentiated.

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Interviewer: What about the acceleration of the ball as it is going up? Student X: If they did not want to say acceleration or deceleration they can use signs

but they must state the signs. There was a change in velocity, initial velocity starts from zero, and then it increases, once it stops, the final velocity will be zero again.

Student X: The acceleration of the ball is decreasing as it is going up, because here it becomes zero, it is decreasing.

Student X: Coming back, it is increasing because its speed is increasing. Although there are some elements of correct thinking by the student in that acceleration is zero at maximum height and a change in velocity occurs, however acceleration is NOT decreasing but constant and it is the velocity that decreases uniformly as the object goes upwards (see category D.1. for in-depth discussion). D.1. Algebraic signs for velocity and acceleration assigned as direction and / or magnitude For velocity, the student correctly states that the algebraic sign is assigned for direction.

Interviewer: What does the sign for velocity mean? Student: It might confuse them if I give them the velocity using sign convention

for direction. But for velocity because of its direction I will be using sign convention there. If I am going to use negative or positive, ok, this is for direction.

Interviewer: What about signs for acceleration? Student Z: Its acceleration is directed upwards, so it's plus, when it comes to a

certain point there is no acceleration, it will be zero, and when it is coming down, the acceleration is also a plus, the object is going faster.

The student incorrectly states that the direction of acceleration is upwards for a falling object but allocates a correct (+) sign for upward motion. The student also incorrectly says that at maximum height, the acceleration (g) is zero instead of g = 9,8 being directed downwards. The velocity and NOT acceleration is zero. For upward motion, the sign for acceleration refers to the direction of motion but for downward motion, the (+) sign for acceleration is chosen to mean a change in magnitude. Hence, the algebraic sign for acceleration for falling bodies is assigned both as direction and as magnitude. Of significance, students have difficulty in the concept of acceleration as the rate of change in velocity especially in understanding the motion of falling bodies. At maximum height, the acceleration g = Δv/ Δt = (0 - 9,8)/1s = - 9,8. Here the magnitude of acceleration is 9,8 and (–) sign implies acceleration directed downwards (where initially displacement upwards was selected as (+)). E. 1 Algebraic sign for gravitational acceleration cannot be positive but acceleration for straight-line motion can be. This category implies that students assign only a negative sign for acceleration due to a falling body in its downward motion. Thus the notion of an arbitrary allocation of algebraic signs is not fully comprehended.

Interviewer: Can the acceleration of a body thrown up be positive? Student: No, gravitational acceleration can't be positive but normal acceleration

can when you are talking about straight-line motion.

The possibility that a positive sign could be assigned for acceleration for downward motion as well may not be reinforced in school physics – possibly because learners are not explicitly taught about the arbitrary choice of sign convention.

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E.2 Use algebraic signs of the number line as directions for displacement. The mathematical drawing of a number line, a positive (+) integer to the right of the zero origin and a negative (–) integer to the left is often interpreted incorrectly as directions for displacement. This notion is different from B.1. The zero origin here is explicitly specified and a single integer is incorrectly used to specify the displacement instead of the position of the object. This is an incorrect conception of interpreting signs for displacement as the definition of displacement is the change in position and the sign resulting from this change gives the direction of the object and hence a sign for displacement. Arons (1997) states that the concepts of position and displacement must be first firmly established before more difficult vector concepts are introduced. For example, the integers +5 and +10 are position points on a number line. The change in position = (+ 10) – (+5) = +5 indicates motion to the right if the + sign is arbitrarily chosen as motion to the right. If the change in position = (+5) – (+10) = –5, the displacement is to the left. E.3 Algebraic signs applied as directions to vector-kinematical concepts This category corresponds to the norm in physics of assigning algebraic signs to vector-kinematical concepts of displacement, velocity and acceleration.

Interviewer: Where did you use + and – signs? Student: In science, it had to do with the direction…a bit confusing.

The student understands that for one-dimensional motion, signs are used to express direction of motion of object. From Grade 11 (16-17-years-old), students are taught for the first time, in school physics, that a vector has magnitude and direction and that signs (+) and (–) can be interpreted as directions, for example, a positive sign (+) means the object is moving to the right and a negative sign (–) means its moving to the left and the number, its magnitude. The conceptual shift from the early understanding of signs as only as positive and negative 'mathematical' numbers with magnitude only in Grade 9 and 10 mathematics was possibly confusing to students at the first introduction of signs as representing directions in Grade 11.

Interviewer: Would you use any signs for acceleration? Student: There will be a sign because the ball is moving in an opposite direction.

Acceleration will be negative because it will be moving downwards. Interviewer: Why? How come? Student: Gravity acted on it, acceleration going up is positive so going down will

be negative. For this student, the sign for acceleration is taken to be the same as the sign allocated for the direction in which the object is moving. For example, if the object is moving up, it is allocated a (+) sign for the direction of displacement and a (+) sign for the direction of acceleration as well. The sign for acceleration as representing its direction is often incorrectly interpreted together with the initial sign chosen for displacement. If initially, the upward displacement of the object is chosen as (+) then the sign for acceleration is (–) implying that the acceleration is directed downwards. What needs to be understood conceptually is the acceleration of a falling body is always directed downwards irrespective whether the object is moving up or down. Hence, if (–) sign is chosen for upward displacement then (+) sign representing the downward direction of acceleration should be chosen and understood. Discussion Phenomenographic analysis of this study (see Table 2) has revealed five core understandings of algebraic sign convention that physics students hold. Two broad categories of algebraic sign convention in the form of integer signs can be discerned from Table 1, algebraic signs are conceptualised in terms of the number-line system used in Mathematics, namely, as algebraic

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signs (+ and –) is used in the context of understanding integers as positive and negative numbers as having magnitude (category B1) and as having directions in Physics (category E3). The study reveals that these student-teachers hold a mix of algebraic sign conventions that are inherently linked to their vector-kinematical conceptual understanding. In particular, the concepts of scalars and vectors and their associated signs are not clearly differentiated. Scalars are physical quantities having magnitude only whereas vectors have both magnitude and direction. The vector-kinematical concepts of displacement, velocity and acceleration are vectors and are connected by their definition in physics; hence the algebraic signs, used for direction of motion, have consistent application in their meaning when solving vector-kinematical problems. In physics, the quantities are defined as follows: displacement is the change in position; velocity is the rate of change of displacement, and acceleration is the rate of change of velocity. To illustrate the confusion in the interpretation of signs for concepts of velocity and acceleration, consider the example of a car speeding up from 5 m.s-1 to 10 m.s-1 in 5 s. Let us arbitrarily choose the direction of displacement to the right as (–), then the direction of velocity to the right is also (–). The acceleration is a = Δv/ Δt = [(-10) – (-5)]/5s = –1 m.s-2. This is interpreted as the magnitude of acceleration of 1 m.s-2 and the (–) sign as the direction of acceleration to the right. Students often interpret the (–) sign of acceleration together with its magnitude in most cases to be 'slowing down' without examining their choice of sign convention. If the car was slowing down from 10 m.s-1 to 5 m.s-1 in 5s, and if displacement to right is (+) sign, then a = [(+5) – (+10)]/5s = –1 m.s-2. In this case the direction of acceleration is to the left. Hence the algebraic sign for displacement, velocity and acceleration is to be interpreted as a direction only. Some physic textbooks discuss different ways of interpreting signs that may cause some confusion to students. Numerous incorrect preconceptions, particularly with regard to the role of the component of the velocity vector have already been observed among physics students (Aquirre, 1988). The special case of allocating positive and negative signs for vectors in one-dimensional motion confuses them with scalars, which can often have two signs (e.g., charge, potential energy, total energy and time). Bauman (1992) raises an important point in this area of research that we often do not explain to the novice that we freely shift between vectors and their components. Many students, as well as school teachers, are not, however, aware of this important distinction, possibly because only one-dimensional vector-kinematical motion is taught for Grade 11 and 12 learners in South African schools. Warren (1997) believes that most difficulties are caused by the introduction of only a one-dimension analysis of motion in physics. He suggests that it is conceptually better to start with motion in a plane, i.e. in two dimensions and in this way the essential nature of a vector, having magnitude and direction, is explained at the beginning of vector-kinematics. When students study tertiary physics, algebraic signs must now be interpreted as scalars in one, two and three-dimensional Cartesian co-ordinate system so that complex physics problems can be solved efficiently. Within the context of physics, one-dimensional motion can be interpreted as a special case of motion where algebraic signs understood to be scalar is now interpreted as direction for displacement, velocity and acceleration The notation of vector symbols must also be emphasised in the study of vector-kinematics. Students should be encouraged to use arrows, boldface type, or underline the vector symbols. If they do not distinguish between scalar and vector quantities in the notation, they will probably not make the distinction in their thinking either (Arons, 1997). In the case of velocity, some students interpreted the sign for velocity as a change in magnitude rather than in direction (one-dimensional motion only). This perception could possibly stem from the number line system where positive numbers to the right of zero are increasing in

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magnitude and negative numbers to the left of zero are decreasing in magnitude or from the confusion of acceleration and its effects on velocity. Another interpretation is that the concept of addition uses a plus sign that increases the magnitude of a number, while a subtraction uses a minus sign that decreases the magnitude of a number. Thus the sign of velocity as positive is experienced as evidenced by interviews as motion 'faster' and velocity negative as motion 'slower'. Thus we may infer that velocity may not be understood as an instantaneous quantity. The interpretation of 'instantaneous velocity' as a number referring to a single instance is a conceptual 'hurdle' for many students (Rosenquist & McDermott, 1987; Trowbridge & McDermott, 1980). Velocity is not correctly interpreted as how fast the object is moving but rather as 'faster' or 'slower', which is acceleration. In this study, some students, confusing the concept of velocity with acceleration, presented the incorrect notion that velocity meant increase in speed. For falling bodies and incline planes, some students incorrectly stated that as the ball fell, acceleration would increase and, as the ball was falling, it was moving faster and therefore, the acceleration was increasing. Some students experienced an increase in speed as directly proportional to an increase in acceleration. These students thought that the ball when thrown upwards already had acceleration upwards when it started from rest and it continued through top speed. The students did not realise that a constant force acted on the object and, hence, a constant acceleration was produced in a downward direction even though the ball was up going and coming down. Some students, who knew that acceleration due to gravity (g) always acts downwards, incorrectly insisted that the sign of g is always negative. Students in this study do not seem to be aware that the choice of a coordinate axis is itself arbitrary. The analysis of motion of a point can be in any arbitrary direction. We can always consider the motion along definite straight lines, that is, along the coordinate axis. Upward direction can be positive or negative and the choice of coordinate axes dictated by the conditions of the problem and that g can be allocated a positive sign if displacement downward was originally taken as positive. For falling bodies, students in this study said that if a particle's velocity is zero then its acceleration is also zero at maximum height. A possible reason for student teachers' responses is that they memorize rules and definitions without discrimination and often retrieve and remember knowledge in fragments without considering the context (Rief & Allen, 1987). This study, and other research, confirms that students bring from their schooling experience to the formal study of physics, naïve and mixed understandings of the common concepts associated with motion in the study of vector-kinematics. The concepts of position, displacement, speed, velocity and acceleration and their associated algebraic sign convention are not distinguished clearly, not applied to real life motion and often exist as memorised definitions used mainly for tests and examinations and thus may soon be forgotten. Implications for teaching This study suggests that algebraic signs as a sign convention and concepts in vector-kinematics which are central to students' scientific understanding in a wide range of topics should be analysed more in-depth and be given an appropriate amount of teaching time. This can be achieved by first exploring student teachers' alternative conceptions of algebraic sign convention in vector-kinematics, and then, by engaging student teachers' perceptions, using the outcome space for sign convention in vector-kinematics invented in this study (Table 2). Furthermore, the introduction of motion in two and three dimensions first, and then simplifying for one-dimension of motion would provide a better and deeper understanding of vector-kinematics. Conclusion The results of the study, for sign convention in vector-kinematics, conclude that there are five qualitatively different ways of conceptualising sign convention in vector-kinematics (Table 2). The results of the study also concludes, for everyday understandings of algebraic sign

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convention, sign convention in the form of integer signs are experienced in terms of the number-line system used in Mathematics, and, in Science, to indicate direction of motion of an object (Table 1). The study also confirms that students hold a mix of algebraic sign conventions, which is compounded with poor understanding of vector-kinematics concepts. Furthermore, the study reveals that students, on their own, fail to formalise a consistent idea of algebraic sign convention in vector-kinematics. Acknowledgements This paper is based on work supported by the National Research Foundation towards a PhD obtained from the University of Western-Cape (UWC). I acknowledge the assistance of Prof. Cedric Linder in this regard. Any opinion, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Research Foundation or the UWC. References Allie, S., & Buffler, A. (1998). A course in tools and procedures for Physics 1. American Journal of

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