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Middle School Students’ Learning Progressions for Scientific Modeling Force and Motion

Daesung Bae

Kunja Middle School, Siheung 429-410

Junehee Yoo ^∗

Department of Physics Education, Seoul National University, Seoul 151-742 (Received 7 June 2012 : revised 13 June 2012 : accepted 30 July 2012)

The purposes of this study are to investigate middle school students’ learning progressions for modeling force and motion phenomena and to identify students’ difficulties by analyzing Korean middle school students’ problem solving involving a forces and motion phenomenon in a everyday context. A test item based on previous research was developed. The participants were 59 high achieving of middle school students. Students’ responses were rated and analyzed by using the developed assessment framework and rubric according to the Modeling Schemata of Halloun. Mid- dle school students’ learning progressions for modeling were classified into five levels by analyzing

-809-

item characteristic curves and the distribution of students by levels. Specific features of each level and difficulties in modeling were analyzed quantitatively and qualitatively. Middle school students’

learning progressions for force and motion were classified into “Identification of Model Composi- tion”, “Concept Representations”, “Modeling with Some Concept Organization”, “Modeling with Scientific Concept Organization,” and Modeling with Scientific Model Structure”. Students seemed to have difficulties in identification of invisible model composition except the concept of speed and represented them as idealized concepts. Also, the causal facets of model structure and concept organization appeared as students’ difficulties, which is thought to be caused by students’ ideas about force and motion in a conceptual world.

PACS numbers: 01.40.Ej

Keywords: Scientific model, Modeling, Learning Progression, Force and Motion, Difficulty

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∗

E-mail: [email protected]

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Table 1. A learning progression for understanding models as generative tools for predicting and explaining (Schwarz, 2009).

Level Performances

4 Students construct and use models spontaneously in a range of domains to help their own thinking.

Students consider how the world could behave according to various models. Students construct and use models to generate new questions about the behavior or existence of phenomena.

3 Students construct and use multiple models to explain and predict more aspects of a group of related phenomena.

Students view models as tools that can support their thinking about existing and new phenomena.

Students consider alternatives in constructing models based on analyses of the different advantages and weakness for explaining and predicting these alternative models possess.

2 Students construct and use a model to illustrate and explain how a phenomenon occurs, consistent with the evidence about the phenomenon.

Students view models as a means of communicating their understanding of a phenomenon rather than a tool to support their own thinking.

1 Students construct and use models that show literal illustrations of a single phenomenon.

Students do not view a model as tool to generate new knowledge, but do see models as a means of showing others what the phenomenon looks like.



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¸+ þ A`  ¦ ½ ¨^ ‰& h Ü ¼– Ð › ' a ¹ 1 Ͻ + É Ã º e ”   H ‰ & ³ © œõ  Æ Ò © œ& h “   s 



: r`  ¦ ƒ       H [ O " î • ¸½ ¨– Ð & ñ _  % i  .

Halloun [23]“ É r Ó ü t o & h  ‰ & ³ © œ\ " f
 
  H : £ ¤& ñ ô  Ç Ä » + þ

A`  ¦ ³ ð © œ l  0 AK " f Ó ü t o & h  > _  ½ ¨› ¸
 ' Ÿ 1 l x \ " f
 


  H : £ ¤& ñ ô  Ç Ä »+ þ Aõ  ƒ  >   ) a > h¥ Æ & h  ^ ‰> \  ¦ õ † < Æ& h  — ¸ + þ

As  “ ¦ % i  . ¢ ¸ô  Ç — ¸+ þ A½ ¨$ í õ & ñ `  ¦ [ O " î l  0 A 

#

Œ — ¸+ þ A½ ¨$ í • ¸d ” (Modeling Schemata)`  ¦ ] jr  % i   H X

<, — ¸+ þ A½ ¨$ í • ¸d ” “ É r — ¸+ þ A• ¸d ” (Model Schema)õ  > h¥ Æ 

•

¸d ” (Concept Schema)Ü ¼– Ð [ jì  r  o  ) a   [23]. — ¸+ þ A• ¸d ”  õ

 > h¥ Æ • ¸d ” “ É r y Œ •y Œ • 4> h_  " é ¶ Ü ¼– Ð ½ ¨$ í ÷ &  H X <, — ¸+ þ A

•

¸d ” “ É r — ¸+ þ A% ò % i (Domain), — ¸+ þ Aכ ¹™ è(Composition), — ¸ + þ

A½ ¨› ¸(Structure) x 9  — ¸+ þ A› ¸f ”  o(Organization)_ 

" é ¶ Ü ¼– Ð. > h¥ Æ • ¸d ” “ É r > h¥ Æ # 3 0 A(Scope), > h¥ Æ ³ ð

‰

&

³(Expression), > h¥ Æ › ¸f ”  o(Organization), x 9  > h¥ Æ  Ã

º| ¾ Ó o(Quantification)_  " é ¶ Ü ¼– Ð ½ ¨$ í  ) a  “ ¦ ] jr  

%

i   [23]. Halloun [23]“ É r Hestenes [2] õ  ² ú ˜o  — ¸+ þ A½ ¨$ í

•

¸d ” `  ¦ : Ÿ x ô  Ç — ¸+ þ A½ ¨$ í õ & ñ `  ¦ Šҁ © œ % i t ë ß –,   H‘ : r& h Ü ¼

–

Ð — ¸+ þ A½ ¨$ í õ & ñ “ É r Hestenes [2] s  ] jr ô  Ç — ¸+ þ A½ ¨$ í õ 

&

ñ õ  ° ú   . Halloun [23]s  ] jr ô  Ç — ¸+ þ Aכ ¹™ è  H @ / © œõ  1 l x

“

 , “ ¦Ä » > h¥ Æ ,  © œI  > h¥ Æ ,  © œ  ñ Œ •6   x > h¥ Æ  1 p x“  X <, s \  ¦ Æ

ÒØ  ¦   H  כ s  Hestenes [2]s  ] jr ô  Ç — ¸+ þ A_  l Õ ü t é ß –>  s

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:õ   © œ  ñ Œ •6   x Z O g Ë :`  ¦ & h 6   x # Œ — ¸+ þ A_  כ ¹™ è[ þ t`  ¦ › ¸f ” 



o # Œ œ íl  — ¸+ þ A`  ¦ ½ ¨$ í   H é ß –> s  . — ¸+ þ A½ ¨$ í • ¸d ” 

\

" f — ¸+ þ A½ ¨› ¸  H l  † < Æ& h  8 £ ¤€  ,  © œ  ñ Œ •6   x& h  8 £ ¤€  ,  © œ I

& h  8 £ ¤€  , “  õ & h  8 £ ¤€  `  ¦ t   H X < [23], s  ×  æ l  † < Æ

&

h

 8 £ ¤€  “ É r Hestenes [2] s  — ¸+ þ A½ ¨$ í \ " f ] jr ô  Ç ‘ý a³ ð>  [ O

& ñ ’\  K { © œô  Ç . Hestenes [2]_  î  r1 l xZ O g Ë :õ   © œ  ñ Œ •6   x Z O

g Ë :“ É r y Œ •y Œ • Halloun [23]_  — ¸+ þ A½ ¨› ¸ ×  æ  © œI & h  8 £ ¤€   õ

  © œ  ñ Œ •6   x 8 £ ¤€  \  K { © œô  Ç “ ¦ ½ + É Ã º e ”  . î  r1 l x Z O g Ë : õ

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 o  H  © œ  ñ Œ •6   x& h  8 £ ¤€  õ   © œI & h  8 £ ¤€  `  ¦ “  õ & h  8 £ ¤€   Ü

¼– Ð ƒ  >  €  " f s À Ò# Qt  9, + þ A$ í ô  Ç — ¸+ þ A`  ¦ ³ ð‰ & ³½ + É M :

>

h¥ Æ ³ ð‰ & ³õ  > h¥ Æ à º| ¾ Ó o s À Ò# Q”   . Halloun [23]s  ]

jr ô  Ç — ¸+ þ A½ ¨$ í • ¸d ” _ 
 Qt  " é ¶ “   — ¸+ þ A% ò % i , — ¸+ þ A

›

¸f ”  o x 9  > h¥ Æ # 3 0 A  H y Œ •y Œ •_  s  : r î ß –\ " f — ¸+ þ A_  # 3 0 A,

—

¸+ þ A[ þ t  s _  › ' a >  x 9  > h¥ Æ _  0 Au \  ¦ & ñ K Šҍ  H % i ½ + É`  ¦ ô

 Ç .

Schwarz 1 p x [10]“ É r õ † < Æ& h  — ¸+ þ A`  ¦ õ † < Æ& h  ‰ & ³ © œ`  ¦ [ O " î

“ ¦ \ V8 £ ¤ l  0 AK " f Å Òכ ¹ô  Ç : £ ¤f ç \  œ í& h `  ¦ ´ ú » ¡ § Ü ¼– Ð+ ‹

>

\  ¦ Æ Ò © œ o “ ¦ é ß –í  H  o   H ³ ð © œs  “ ¦ & ñ _  “ ¦ œ í1 p x

†

< ÆÒ q t_  Õ ªa Ë >  Ò q t$ í — ¸+ þ A½ ¨$ í õ & ñ `  ¦ ì  r$ 3  # Œ — ¸+ þ A½ ¨$ í Ã

ºï  r`  ¦ Table 1 õ  ° ú  s  4à ºï  r Ü ¼– Ð ] jr  % i  . Schwarz 1

p x [10]“ É r s  Qô  Ç   õ – РÒ'  † < ÆÒ q ts  ½ ¨$ í ô  Ç — ¸+ þ As  l  Õ

ü t& h  — ¸+ þ A\ " f [ O " î & h  — ¸+ þ AÜ ¼– Ð, ‰ & ³ © œ\  @ /ô  Ç [ O " î s  ˜ Ð



  8 & ñ “ § o÷ &  H € ª œ © œÜ ¼– Ð — ¸+ þ A½ ¨$ í à ºï  r s  ”  „  ô  Ç 



 H  כ `  ¦ S X ‰ “  ½ + É Ã º e ”  .

s

 © œõ  ° ú  “ É r ‚  ' Ÿ ƒ  ½ ¨\  ¦  „ ½ ÓÜ ¼– Ð ‘ : r ƒ  ½ ¨\ " f  H õ † < Æ

&

h  — ¸+ þ A`  ¦ z  ´] j [ j> \ " f { 9 # Q
  H Ó ü t o & h  ‰ & ³ © œ\ 
  è

ß – Ó ü t o & h  z  ´^ ‰ü <  |  x 9  Õ ª כ [ þ t _  : £ ¤$ í `  ¦ Æ Ò © œ& h “   > h

Fig. 3. (Color online) Model and modeling for the car problem.

Table 2. Analytic Framework for Evaluation of modeling.

Dimension Category Brief Operational Definition Model Composition Object representation How to represent the object Agent representation How to represent the agent

Intrinsic Concept representation How to represent a characteristic property of the object State Concept representation How to represent a property that characterizes

the behavior of the object

Interaction Concept representation How to represent mutual actions between the object and the agent Model Structure Topology Facet Set the object in an appropriate coordinate system

Interaction Facet Express the interaction concepts primarily with interaction laws State Facet Describe how the object behave in a specific reference system Causal Facet Explain why the objects behave the way they do in a specific

reference system

Concept Expression Verbal expression Express the concept verbally Symbolic expression Express the concept symbolically

Iconic expression Express the concept iconically Mathematical expression Express the concept mathematically

Concept Organization Relation among the concepts Set relation among intrinsic, state and interaction concepts

¥ Æ

 [ j> \  e ”   H õ † < Æ& h  > h¥ Æ Ü ¼– Ð ³ ð © œ “ ¦ Õ ª > h¥ Æ [ þ t _ 

›

¸f ”  oü < ³ ð‰ & ³`  ¦ : Ÿ x K " f 3 l q ³ ð“   Ó ü t o & h  ‰ & ³ © œ`  ¦ l Õ ü t·[ O 

"

î “ ¦ \ V8 £ ¤   H > h¥ Æ ^ ‰> – Ð & ñ _  % i  . — ¸+ þ A½ ¨$ í “ É r  

’

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&

h

 ‰ & ³ © œ\  & h ] X ô  Ç — ¸+ þ A_  כ ¹™ è\  ¦ Æ ÒØ  ¦ “ ¦ › ¸f ”  # Œ — ¸ + þ

A_  ½ ¨› ¸\  ¦ ë ß –[ þ t “ ¦ & h 6   x·¨ î    H “  t õ & ñ s  “ ¦ ½ + É Ã

º e ”  . ‘ : r ƒ  ½ ¨\ " f s K ô  Ç — ¸+ þ A x 9  — ¸+ þ A½ ¨$ í _  • ¸d ” 

`

 ¦  Ž   ë  H † ½ Ó \ Vr – Ð # Œ Fig. 3\  ] jr  % i  .

Figure 3 \ " f ] jr   ) a  ü < ° ú  s  z  ´] j [ j> \ " f { 9 # Q




 H Ó ü t o & h  ‰ & ³ © œ\   H Ó ü t o & h  z  ´^ ‰“   @ / © œõ  1 l x“  s  ” > r F  ô

 Ç . @ / © œ x 9  1 l x“  õ  @ / © œ_  ½ ¨› ¸& h  : £ ¤$ í , ' Ÿ 1 l x& h  : £ ¤$ í ,

@

/ © œõ  1 l x“  _   © œ  ñ Œ •6   x`  ¦ — ¸+ þ A_  כ ¹™ è– Ð “  t  # Œ s 

Fig. 4. (Color online) Test Item.



© œ o “ ¦ > h¥ Æ  [ j> \ " f s \  K { © œ   H > h¥ Æ `  ¦ ‚  & ñ 

#

Œ & ñ ’  — ¸+ þ AÜ ¼– Ð ³ ð © œô  Ç . ³ ð © œ  ) a > h¥ Æ [ þ t`  ¦ " f– Ð › ¸f ” 



o # Œ — ¸+ þ A_  ½ ¨› ¸\  ¦ › ¸w n  “ ¦ s \  ¦  „ ½ ÓÜ ¼– Ð 3 l q ³ ð“   Ó

ü

t o & h  ‰ & ³ © œ`  ¦ l Õ ü t·[ O " î ·\ V8 £ ¤ ½ + É Ã º e ”   H > h¥ Æ ^ ‰> \  ¦ & ñ

’

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—

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“

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\

 ¦ : Ÿ x ô  Ç — ¸+ þ A½ ¨› ¸_  ½ ¨$ í 1 p x _  " é ¶ Ü ¼– Ð s À Ò# Q”   “ ¦

‘

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\

 @ /ô  Ç & ñ _   H Halloun [23] _  — ¸+ þ A½ ¨$ í • ¸d ” \ " f ] jr  ô

 Ç  \  ¦  | Ã Û .

‘

: r ƒ  ½ ¨\  ‚ à Ð# Œô  Ç † < ÆÒ q t[ þ t s  K { © œ “ §¹ ¢ ¤ õ & ñ \ " f Ó ü t o  Z O

g Ë :_  & ñ | ¾ Ó& h  ³ ð‰ & ³\  @ /ô  Ç † < Æ_ þ v  â + « >s  \ O   H  כ Ü ¼– Ð ó ø Í é

ß – # Œ — ¸+ þ A ½ ¨$ í _  " é ¶ ×  æ > h¥ Æ à º| ¾ Ó o  H ¨ î  כ ¹™ è\ 

"

f ] jü @ % i  . Õ ª   õ  Table 2ü < ° ú  s  — ¸+ þ A½ ¨$ í 0 p x§ 4 \ 

@

/ô  Ç ¨ î d  ¦`  ¦ ½ ¨$ í % i  .

Figure 3 \ " f ] jr   ) a  ü < ° ú  s  † < ÆÒ q t[ þ t s  t “ ¦ e ” 



 H > h¥ Æ  [ j> _  > h¥ Æ  ^ ‰>   H — ¸+ þ A½ ¨$ í \ " f ×  æ כ ¹ô  Ç % i ½ + É

`

 ¦ ô  Ç . Õ ª Q
 j Ë µõ  î  r1 l x % ò % i \  @ /ô  Ç ‚  ' Ÿ ƒ  ½ ¨\  _  

€

 , @ / à º_  † < ÆÒ q t[ þ t“ É r ¹ ¡ §f ” s   H Ó ü t ^ ‰\   H j Ë µs   Œ •6   x 

“

¦ ¹ ¡ §f ” s t  · ú §  H Ó ü t ^ ‰\   H j Ë µs   Œ •6   x “ ¦ e ” t  · ú § “ ¦ Ò q

ty Œ •  9, Ó ü t ^ ‰_  î  r1 l x“ É r  Œ •6   x   H j Ë µ_  ß ¼l \  q Y V 

“

¦, { 9 & ñ ô  Ç j Ë µs   Œ •6   x €   5 Å q§ 4 s  { 9 & ñ “ ¦, ½ + ˧ 4 s  0s 

€

  Ó ü t ^ ‰  H Ö ¼ 9”     H 1 p x  € ª œô  Ç Ò q ty Œ •`  ¦ t “ ¦ e ”   H  כ Ü

¼– Ð ˜ Г ¦÷ &% 3   [3–6].

s

 Qô  Ç † < ÆÒ q t_  Ò q ty Œ •\  @ /ô  Ç ƒ  ½ ¨  H ë  H † ½ Óì ø Í6 £ x s  : r s  • ¸ { 9

H † d \     à ºï  r _   © œ[ j o\  @ /ô  Ç ƒ  ½ ¨– Ð    o “ ¦ e ” 



. \ V\  ¦ [ þ t # Q Alonzo 1 p x [11]“ É r 7 † < Ƹ  \ " f 9† < Ƹ   † < ÆÒ q t[ þ t _

 j Ë µõ  î  r1 l x > h¥ Æ _  à ºï  r _   © œ[ j o\  ¦ 5 t  à ºï  r Ü ¼– Ð

 Í Ç x  H X <, à ºï  r0“ É r Å Ò] j\ " f # Á # Qè ß – 6 £ x ² ú š`  ¦ ô  Ç  â Ä º s

 9, à ºï  r1“ É r j Ë µ`  ¦ ¶ ú ˜ e ”   H Ò q tÓ ü t ^ ‰\  _ K  x 9  
 { © œ

" f µ 1 ÏÒ q t   H  כ Ü ¼– Ð Ò q ty Œ •   H  â Ä º, à ºï  r2  H î  r1 l x   H Ó

ü

t ^ ‰\  ° ú  “ É r ~ ½ ӆ ¾ ÓÜ ¼– Ð  Œ •6   x   H j Ë µs  ” > r F ô  Ç “ ¦ Ò q ty Œ •

  H  â Ä º, à ºï  r3“ É r Ó ü t ^ ‰\   Œ •6   x   H j Ë µs  \ O  
 ½ + ˧ 4  s

 0“    â Ä º\  Ó ü t ^ ‰  H & ñ t K  e ”  “ ¦ Ò q ty Œ •   H  â Ä º, à º ï

 r4  H Ó ü t ^ ‰\   Œ •6   x   H j Ë µ“ É r Õ ª ß ¼l \  q Y V   H 5 Å q • ¸

\

 ¦ µ 1 ÏÒ q tr †   “ ¦ Ò q ty Œ •   H  â Ä ºs  .

III. Ž ì ŏ Œ8 ý U ê s0 n É õ m Í ± n Ç 

1. Ž ì ŏ Œ8 ý 6 V ê s

‘

: r ƒ  ½ ¨  H " fÖ  ¦ r ? / ™ èF  ×  æ † < Ɠ § † < ÆÒ q t ×  æ — ¸ @ /† < Ɠ § % ò F

“ §¹ ¢ ¤" é ¶ \   m   H õ † < Æ  © œ0 A Ý ¶ 1, 2 † < Ƹ   † < ÆÒ q t 59" î `  ¦ @ /

Table 3. The Rubrics

Dimension Category Variable Score

Model Object (Car) As a point 2

Composition As an object with dimensions 1

Not classifiable 0

Agent (Road) Road affect motions of a car 2

Brake affect motions of a car 1

Not classifiable 0

Intrinsic Concept Mass 2

(Mass) Representation Use everyday expressions such as “becoming lighter” rather than “mass” 1

Not classifiable 0

State Concept Speed 2

(speed) Representation Use everyday expressions such as “slow” rather than “speed” 1

Not classifiable 0

Interaction Concept Frictional force 2

(Frictional force) Use “brake” instead of “frictional force” 1

Representation Not classifiable 0

Model Topology Facet Use appropriate coordinate system with distance 2 Structure (coordinate system) Use inappropriate coordinate system without distance 1

Not classifiable 0

Interaction Facet Frictional coefficient & weight 2

(Interaction law) One of them 1

Not classifiable 0

State Facet “Speed is reduced uniformly” 2

(describe behavior) “Speed is reduced” without expressions of “uniformly” 1

Not classifiable 0

Causal Facet “Speed is reduced by a force of opposite direction” 2 (explain behavior) “Speed is reduced by certain action” 1

Not classifiable 0

Concept Intrinsic Concept Express mass verbally, symbolically and mathematically 2 Expression (Mass) expression Express mass verbally and symbolically/mathematically 1

Not classifiable 0

State Concept Express mass verbally, symbolically, iconically and mathematically 2 (speed) Expression Express mass verbally and symbolically/iconically and mathematically 1

Not classifiable 0

Interaction Concept Express mass verbally, symbolically and iconically, mathematically 2 (Frictional force) Express mass verbally, symbolically/iconically, mathematically 1

expression Not classifiable 0

Concept Relation Between “Speed changes are in inverse proportion to mass” 2 Organization Mass & speed Unclear relation between speed change and mass 1

Not classifiable 0

Relation Between “Speed changes are in inverse proportion to force” 2 force & speed Unclear relation between speed change and force 1

Not classifiable 0



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×

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`

 ¦ † < Æ_ þ v % i  “ ¦ & ñ % i  . “ ¦1 p x † < ÆÒ q tõ  @ /† < ÆÒ q t`  ¦ @ /



© œÜ ¼– Ð ô  Ç ‚  ' Ÿ ƒ  ½ ¨\  _  €   [19], ¿ º | 9 é ß –\ " f — ¸+ þ A½ ¨

$ í

0 p x§ 4 “ É r Ä »_ p ô  Ç s \  ¦
 ? /t  · ú §€ Œ ¤ . ‘ : r ƒ  ½ ¨\ 

"

f• ¸ 1 l x{ 9 ô  Ç “ §¹ ¢ ¤ õ & ñ `  ¦ s à ºô  Ç ¿ º > h † < Ƹ  _  † < ÆÒ q t[ þ t s 

ƒ

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#

Q — ¸+ þ A½ ¨$ í 0 p x§ 4 _   € ª œ o\  • ¸¹ ¡ § s  ÷ &>  % i  .

Table 4. Score Result.

Dimension Category Scores(n=59) Mean SD Difficulty Discrimination Correlation Coefficient

2 1 0 with sum of scores

Model Object representation 1 43 15 0.76 0.47 0.381 0.233 .364

Composition Agent representation 7 35 17 0.83 0.62 0.415 0.167 .210 Intrinsic Concept representation 5 1 53 0.19 0.57 0.093 0.267 .498

State Concept representation 49 2 8 1.69 0.70 0.847 0.467 .498

Interaction Concept representation 23 10 26 0.95 0.92 0.475 0.767 .647

Sum(10) 4.42 1.71

Model Topology Facet 9 43 7 1.03 0.52 0.517 0.233 .413

Structure Interaction Facet 1 1 57 0.05 0.29 0.025 0.067 .358

State Facet 11 31 17 0.90 0.69 0.449 0.567 .618

Causal Facet 13 16 30 0.71 0.81 0.356 0.633 .548

Sum(8) 2.69 1.50

Concept Intrinsic Concept expression 0 5 54 0.08 0.28 0.042 0.133 .523

Expression State Concept expression 12 35 12 1.00 0.64 0.500 0.533 .587

Interaction Concept expression 9 17 33 0.59 0.75 0.297 0.600 .656

Sum(6) 1.68 1.18

Concept Relation between Mass & speed 3 2 54 0.14 0.47 0.068 0.233 .531

Organiztion Relation between Force & speed 5 16 38 0.44 0.65 0.220 0.600 .760

Sum(4) 0.58 0.97

Sum(28) 9.37 4.48 0.335 0.393

Cronbach α = .784

2. ß e È à U Ø]  § õ m Í  z » • ¤ù m É U ê s0 n É

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`

 ¦ ¨ î  l  0 A # Œ Lopes 1 p x [7] s  > hµ 1 Ï # Œ z  ´r ô  Ç — ¸ + þ

A½ ¨$ í 0 p x§ 4   Ž  \ " f  6   x ô  Ç ë  H † ½ Ó ×  æ ô  Dz D G ×  æ † < Ɠ § “ §¹ ¢ ¤ õ

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\

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Table 5. Interpretation of Item Difficulty and Item Discrimination.

Item Difficulty Interpretation Category

> 0.75 easy State Concept representation

0.25∼0.75 middle Object representation, Agent representation, Interaction Concept representation, Topology Facet, State Facet, Causal Facet, State Concept expression,

Interaction Concept expression

< 0.25 hard Intrinsic Concept representation, Interaction Facet, Intrinsic Concept expression, Relation between Mass & speed, Relation between Force & speed

Item Discrimination Interpretation Category

> 0.40 high State Concept representation, Interaction Concept representation, State Facet, Causal Facet, State Concept expression, Interaction Concept expression, Relation between Force & speed

0.30∼0.39 moderate

0.20∼0.29 low Object representation, Intrinsic Concept representation, Topology Facet, Relation between Mass & speed

< 0.19 poor Agent representation, Intrinsic Concept expression,

< 0.10 None Interaction Facet

Table 6. Interpretation of Item Discrimination.

Item Discrimination Interpretation Category

State Concept representation, Interaction Concept representation, State Facet,

> 0.40 high Causal Facet, State Concept expression, Interaction Concept expression, Relation between Force & speed

0.30 ∼ 0.39 moderate

0.20 ∼ 0.29 low Object representation, Intrinsic Concept representation, Topology Facet, Relation between Mass & speed

< 0.19 poor Agent representation, Intrinsic Concept expression,

< 0.10 None Interaction Facet

Table 7. Distribution of Students.

Levels

1 2 3 4 5

Number of 12 24 17 5 1

Students (20%) (41%) (29%) (8%) (2%)

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ºï  r`  ¦ 5 t  à ºï  r Ü ¼– Ð ½ ¨ì  r   H  כ s   { © œ  “ ¦ ó ø Íé ß –

% i  .

Table 8. Mean Scores of Levels.

Dimension Category Levels

5 4 3 2 1

Model Object representation 1.00 1.00 0.86 0.79 0.58

Composition Agent representation 1.00 0.60 1.43 0.63 0.92

Intrinsic Concept representation 2.00 1.20 0.00 0.13 0.00

State Concept representation 2.00 2.00 2.00 1.92 0.67

Interaction Concept representation 1.00 1.80 1.71 0.13 1.08

Sum 7.00 6.60 6.00 3.60 3.25

Model Topology Facet 2.00 1.20 1.29 1.00 1.00

Structure Interaction Facet 2.00 0.00 0.00 0.00 0.08

State Facet 2.00 1.40 1.43 1.00 0.00

Causal Facet 1.00 1.20 1.71 0.33 0.17

Sum 7.00 3.80 4.43 2.33 1.25

Concept Intrinsic Concept expression 1.00 0.60 0.00 0.04 0.00

Expression State Concept expression 2.00 1.40 1.14 1.08 0.25

Interaction Concept expression 2.00 1.60 1.00 0.00 0.75

Sum 5.00 3.60 2.14 1.12 1.00

Concept Relation between Mass & speed 2.00 1.00 0.00 0.04 0.00

Organization Relation between Force & speed 2.00 1.40 1.14 0.00 0.08

Sum 4.00 2.40 1.14 0.04 0.08

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Fig. 5. (Color online) ICC of levels and Distribution of Students according to Levels.

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