國小高年級學童分數加減法之概念探討─S-P表及加權多元計分IRS整合分析

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(1)

(2) . .

(3) S-P IRS .

(4) . .

(5) S-P (S-P chart)

(6) (caution index of problem , CP )

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(10) Abstract The purpose of the study is to integrate the caution index for problem (CP) and the caution index of student (CS) from S-P chart and weighted polytomous item relational structure on knowledge structure analysis of fraction addition and subtraction for fifth and sixth graders. According to the related studies, the researchers conclude four attributes within ten concepts of the fraction addition and subtraction and design the concept test of 16 items respectively. The subjects are 934 fifth graders and 724 sixth graders. Firstly, the S-P chart could provide CP and CS from students’ response data and we could diagnose all items and students. Secondly, we transform dichotomous data into polytomous data by calculating matrix. The polytomous item relational structure could provide the knowledge structures for all types of students. The results of study are as follows. (1) The students with high CS have few relations among the concepts of fraction addition and subtraction so they have fewer strategies to solve problems. (2) The students with low CS have many relations between the concepts of fraction addition and subtraction so they have more strategies to solve problems. (3) The items with the same denominator and the proper fraction are the basic concepts of the fraction addition and subtraction; the items with sub-multiple denominators is difficult concept of the fraction addition and subtraction. Besides, sixth graders are superior in reduction of the fraction. The results can offer references for cognition diagnosis and recommendations for remedial instruction.. Key words: S-P chart, fraction addition and subtraction, weighted polytomous item relational structure, concept structure. II.

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(14) 2-1 i j Z ¿=À!. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 2-2 i j Z ¿=À!. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3-1 ÁOQRÂGÃÄ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3-2 - f/0QÅ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3-3 .- f/0QÅ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3-4 .-OPJF/0=OÆ QÅ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4-1 ()*+,.-;qr¿ÇÈÉ. ................................. 25. 4-2 (+,-= cd"s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4-3 *+,- cd"s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4-4 (+,.- cd"s. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4-5 *+,.- cd"s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. IV. 65.

(15) m 2-1. cd_m.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. m 2-2. cd_m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. m 2-3. /0m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. m 3-1. ±². . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. m 3-2. µ¶¯ÊË. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. m 4-1. (+,na-/0"#m. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 28. m 4-2. (+, A qr-/0"#m. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30. m 4-3. (+, A ' qr-/0"#m. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32. m 4-4. (+, B qr-/0"#m. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33. m 4-5. (+, B ' qr-/0"#m. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 35. m 4-6. (+, C qr-/0"#m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. m 4-7. (+, C ' qr-/0"#m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38. m 4-8. *+,na-/0"#m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41. m 4-9. *+, A qr-/0"#m.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. 43. 36. m 4-10 *+, A ' qr-/0"#m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 45 m 4-11 *+, B qr-/0"#m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46. m 4-12 *+, B ' qr-/0"#m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 48 m 4-13 *+, C qr-/0"#m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49. m 4-14 *+, C ' qr-/0"#m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51. m 4-15 (+,na.-/0"#m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54. m 4-16 (+, A qr.-/0"#m.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56. m 4-17 (+, A ' qr.-/0"#m. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 m 4-18 (+, B qr.-/0"#m. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 m 4-19 (+, B ' qr.-/0"#m. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 m 4-20 (+, C qr.-/0"#m. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 62. m 4-21 (+, C ' qr.-/0"#m. . .. ................................. 64. m 4-22 *+,na.-/0"#m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67. m 4-23 *+, A qr.-/0"#m. 69. V. .....................................

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(41) Takeya (1980) ~c Bart and Krus (1973) $4 01 (ordering theory) $> ?'ef~®©4 l! (ordering coefficient) `g#'; ETakeya (1980, 1991) 7ef$'}Ú° (dichotomous) '~c2'2l (precondition) q4 lE° $ i G j ;'2[ ( 1 ) G ( 0 ) Á!ÃÆ '¬ 2-1 7 (Lin, Bart & Huang, 2006)E 2-1. i. i G j $Á!` j G 1 0. 1 0. G. p10. p11 p01. p00. p1• p0•. p•1. p• 0. 1 = p11 + p10 + p01 + p00. ~c 2-1 $'Takeya (1980,1991) W i ; j l$ m'4 l! rij* = 1 −. p01 '2¯ p01 } i j [$ ( p•1 )( p0• ). ÃÆ' p•1 } j [$ Æ' p0• } i $ ÆE rij∗ !±²³' i ; j l$m²E()'Takeya (1980, 1991) rij∗ $± (threshold) ; .50' i Û j }4 l¬ (®) rij∗ ≥ .50' i ; j $%& (precondition) ': i Û j 4 l')u rij = 1 'É i → j E (°) rij∗ < .50' i ´} j $%&': i Û j 4 l')u rij = 0 'É¯ i Ö jE. 9.

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(261) (Ñ)¸ 1.. `. cd 2 (ÓÊ) →cd 3 (Ò)cd 4 (Ô) cd 5 (Õ) c d 7 (ÊsÖ×) cd 8 (ÊsØÙ) cd 9 (^Ú) Acd 10 (Û ) ÓÊvÜXcdUÝÞcd(. 2.. L%cd 3 (Ò) →cd 4 (Ô) Acd 7 (ÊsÖ×)Kcd 5 (Õ ) →cd 4 (Ô) cd 9 (^Ú) Acd 10 (Û) vDÓÊ,= ýÑþ=cd(. 3.. BU cd 2 (ÓÊ) ì Ù) Kcd 5 (Õ) ì g@¥UÝÞcdì. cd 3 (Ò) Acd 8 (ÊsØ. cd 10 (¼Û) " ½½. ½g@Ü^þcd. ¬= !"5ªì. #$U[u %&Å'cdsU½g@L ()Î Ûv.02 2 Ù*Í +,U-.=/( 4.. O«¬ I A ST67? = A ' ST67cdsì. [u0. ï A ' ST6712½g@¥ ":cdU[\¶3V4( òóô A ST67U6R":cd½g@ »¼2ÒcdvªèÆ 5ILêcd677XUt 67"D< ó 8{UðñX 6R () A ' ST67U6R ó 12O½g@¥ÏÐ "QUcd½g@ A ST6 7 cdsì. [u0 »¼2ÓÊvÝÞcd ÆI A ST67(. 31.

(262) 1Ê 2ÓÊ 3Ò. 4Ô. 6Êsû[u 7ÊsÖ× 9^Ú. 5Õ. 8ÊsØÙ. 10Û l 4-3 A ' ST67. B ST67 l 4-4 v B ST67=cd]l 2· ¸. (cdst[u) UÅÆ«¬ ÇÈÉ2:Æ (i)·. 1.. (cd½g@) A. `. cd 1 (Ê) Acd 6 (Êsû[u) ¥½g@ (0.75 Á½g @) 3ÎX B ST67? ÊAÊsû[uUcdË ÌUD(. 2.. cd 8 (ÊsØÙ) Acd 10 (Û) Í ¨½g@ (½g@À0.5) -cd9ÏÐ(Lêëcd½g@: 0.5 A 0.75 =s Í% UÏÐ ï S67? ÊsØÙA®ÛUcd34(. 32.

(263) (Ñ)¸ 1.. `. cd 1 (Ê) !ì. cd 5 (Õ) cd 9 (^Ú) cd 10 (Û) . 3ÊvcdUÝÞcd cd 1 (Ê) →cd 9 (^Ú) → cd 10 (Û) 6e ;ÊUcd«4<^ÚUcd ú ^ÚUcd«4<ÛUcd( 2.. cd 3 (Ò) !ì. cd 2 (ÓÊ) cd 4 (Ô) 5 (Õ) . cd 8 (ÊsØÙ) cd 9 (^Ú) cd 10 (Û) cd 3 (Ò ) →cd 2 (ÓÊ) →cd 5 (Õ) Acd 10 (Û) 6e ; ÒUcd«4<ÓÊUcd úÓÊUcd«4<ÕAÛ Ucd(. 1Ê 2ÓÊ 3Ò. 4Ô. 6Êsû[u 7ÊsÖ× 9^Ú. 5Õ. 8ÊsØÙ. 10Û l 4-4 B ST67cd]l. 33.

(264) B ' ST67 l 4-5 v B ' ST67=cd]l 2· A¸. (cdst[u) UÅÆ«¬ ÇÈÉ2:Æ. (i)· 1.. (cd½g@). `. °=cd 1 (Ê) ¥½g@ (0.75Á½g@) L|UcdÍ % ½g@ (0.5Á½g@À0.75) 89? B ' ST67«ÄX Í%ÏÐ(. 2.. BU B ' ST67"cd 1 (Ê) cd 3 (Ò) cd 6 ( Êsû[u) U½g@ B ST67¨ cd 8 (ÊsØÙ) Acd 10 (Û) U½g@ B ST67ñ¥ ÆÓU>m36e ( (Ñ)¸. `. O¬« B ' ST67cds?@t Acd 2 (ÓÊ) ì. cd 5 (Õ. ) L|cdBCDt ST67îïðñ*3Eq( òóô ÊAÊsû[u " B ST67U6 Ró¥ÏÐ ÛAÊsØÙUcdxF£Ucd("îïð ñó »¼2ÊAÒvÝÞcd 6e 2GcdsUì 86 75ÐHUcd6R() B ' ST67U6Ró 12 cd 8 (ÊsØÙ) Acd 10 (Û) U½g@ B ST67ñ¥ " nIUcdcd 1 (Ê) cd 3 (Ò) cd 6 (Êsû[ u) ½g@ B ST67 cds?@t ":cdU[\¶3V 4(. 34.

(265) 1Ê 2ÓÊ 3Ò. 4Ô. 6Êsû[u 7ÊsÖ× 9^Ú. 5Õ. 8ÊsØÙ. 10Û l 4-5 B ' ST67cd]l. C ST67 l 4-6 v C ST67=cd]l 2· ¸. (cd½g@) A. (cdst[u) UÅÆ«¬ ÇÈÉ2:Æ (i)·. `. UcdÍ ¨½g@ (½g@À0.5) 89? C S T67«Ä34( (Ñ)¸ 1.. `. cd 1 (Ê) !ì. cd 2 (ÓÊ) cd 4 (Ô) cd 5 (Õ. ) cd 9 (^Ú) Acd 10 (Û) 3ÊvcdUÝÞc d cd 1 (Ê) →cd 4 (Ô) Acd 9 (^Ú) →cd 5 (Õ 35.

(266) ) 6e ;ÊUcd«4<ÔA^ÚUcd ú -cd«4<ÕUcd(°=J cd 1 (Ê) →cd 9 (^Ú) → cd 10 (Û) 6e ;ÊUcd«4<^ÚUcd ú ^ÚUcd«4<ÛUcd( 2.. cd 3 (Ò) →cd 2 (ÓÊ) cd 4 (Ô) 5 (Õ) cd 8 ( ÊsØÙ) Acd 10 (Û) cd 3 (Ò) →cd 4 (Ô) →c d 5 (Õ) 6e ;ÒUcd«4<ÔUcd ú cd«4<ÕUcd(. 3.. cd 6 (Êsû[u) →cd 2 (ÓÊ)cd 5 (Õ) Acd 10 (Û ) Êsû[uvcdUÝÞcd(. 1Ê 2ÓÊ 3Ò. 4Ô. 6Êsû[u 7ÊsÖ× 9^Ú. 5Õ. 8ÊsØÙ. 10Û l 4-6 C ST67cd]l. 36.

(267) ë C ' ST67 l 4-7 v C ' ST67=cd]l 2· ¸. (cdst[u) UÅÆ«¬ ÇÈÉ2:Æ (i)·. 1.. (cd½g@) A. `. UcdÍ ¨½g@ (½g@À0.5) 89? C ' S T67«Ä34(. 2.. l%:cd=½g@ I C ST67 C ' ST67cd½g@: 0.38~0.20 ½g@ C ST67K%(. 3.. BU C ' ST67"cd 1 (Ê) Acd 6 (Êsû[u) U½g@ C ST67¨7X L|cd½g@ B ST67¥ µ! cd 8 (ÊsØÙ) "]ló½g@?¥ªLêcd ÆÓ U>m36e( (Ñ)¸. 1.. `. cd 3 (Ò) !ì. cd 2 (ÓÊ) Acd 5 (Õ) (cd 3 (Ò. ) →cd 5 (Õ) →cd 10 (Û) 6e ;ÒUcd «4<ÕUcd úcd«4<ÛUcd( 2.. cd 4 (Ô) →cd 9 (^Ú) 6e ;ÔUcd«4<^ ÚUcd(. 3.. L|cds?@LMUt ST67NX4cdsUt 2 OPU6R( òóô C ST67»¼2ÊÒÊsû[ucd. vÝÞcd cdst C ' ST67X Ýe"U6Rðñ LM ) C ' ST67cdst10 L½g@ C ST67¥ 6eQ. 37.

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(272) e¾ S-P 3=l 2-2 67. `l=S ¿?67½g@v¨ (½g@À0.5) % (0.5Á½g @À0.75) ¥ (0.75Á½g@) Â35K)¸. `»¼´:cds[\¶. ^_&' L`ÃÄ iNOPQ l 4-8 vNO67=cd]l 2· ¸. (cdst[u) UÅÆ«¬ ÇÈÉ2:Æ (i)·. 1.. (cd½g@) A. `. cd 8 (ÊsØÙ) Í %½g@ (0.5Á½g@À0.75) 67"ÊsØÙUcdÍ%ÏÐ(. 2.. L|cd¥½g@= 67"UcdË ÌUD( (Ñ)¸. 1.. `. cd 3 (Ò) ì. cd 2 (ÓÊ) Acd 5 (Õ) ÒvÓ. ÊAÕUÝÞcdKcd 3 (Ò) →cd 2 (ÓÊ) →cd 8 (Ê. 40.

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(278) 1Ê 2ÓÊ 3Ò. 4Ô. 6Êsû[u 7ÊsÖ× 9^Ú. 5Õ. 8ÊsØÙ. 10Û l 4-10 A ' ST67cd]l. B ST67 l 4-11 v B ST67=cd]l 2· A¸. (cdst[u) UÅÆ«¬ ÇÈÉ2:Æ. (i)· 1.. (cd½g@). `. cd 1 (Ê) Acd 6 (Êsû[u) ¥½g@ (0.75Á½g @) B ST67? ÊAÊsû[uUc dÍ¥ÏÐ(. 2.. cd 8 (ÊsØÙ) v¨½g@ (½g@À0.5) L|cd½g@% (0.5Á½g@À0.75) (3 B ST67? ÊsØÙ Ucd*dF£ )L|cdÍ%ÏÐ(. 45.

(279) (Ñ)¸ 1.. `. cd 1 (Ê) cd 3 (Ò) cd 6 (Êsû[u) cd 7 ( ÊsÖ×) Acd 9 (^Ú) v7Xcd=ÝÞcd B S T67) B ST67ÝÞcdX efgÄ 67õåÐH XcdvUÝÞcd(. 2.. cd 5 (Õ) !hcd 1 (Ê) cd 2 (ÓÊ) cd 3 (Ò) cd 6 (Êsû[u) cd 7 (ÊsÖ×) Acd 9 (^Ú) ì 6e ;4<ÝÞcd 2OPÕUD<ij(. 3.. cd 3 (Ò) Acd 7 (ÊsÖ×) →cd 2 (ÓÊ) →cd 8 (Ês ØÙ) 6e 54<«OPÊsØÙUD<ij(. 1Ê 2ÓÊ 3Ò. 4Ô. 6Êsû[u 7ÊsÖ× 9^Ú. 5Õ. 8ÊsØÙ. 10Û l 4-11 B ST67cd]l. 46.