ᩘᏛᏛ⩦࡟࠾ࡅࡿᢳ㇟ⓗ౛♧ࡢ㌿⛣ඃ఩ᛶ࡟ࡘ࠸࡚⧴⧭ ᑗ༤ ከ἞ぢᕷ❧໭ᰤᑠᏛᰯ᭶ඖ ᩗ ᒱ㜧኱ᏛOn a transfer advantage of abstract examples in learning mathematics
Masahiro KOKETSU (Gifu Tajimi Hokuei Elementary School)
Takashi TSUKIMOTO (Gifu University)㏆ᖺࡢᩘᏛᩍ⫱◊✲࡟࠾࠸࡚㸪Ꮫ⩦ࡢ㌿⛣࡟ࡣ㸪ලయ౛ࡼࡾࡶᢳ㇟౛ࡢ᪉ࡀ᭷ຠ࡛࠶ࡿࡇ࡜ࡀ♧ࡉࢀ࡚࠸ࡿࠋᮏ◊✲࡛ࡣࠕ࢖ࣛࢫࢺࡢ᭷↓ࠖ࡜ࠕᚠ⎔ⓗᵓ㐀ࡢ᜛ពᛶࡢ᭷↓ࠖࡢ㐪࠸࡟ࡼࡿ㌿⛣ࡢ⏕ࡌࡸࡍࡉࢆ᳨ウࡋࡓࠋࡲࡓ㸪ṇ⟅ᩘࡢ௚࡟㌿⛣ࡢ⏕ࡌࡸࡍࡉࢆ⾲ࡍᣦᶆ࡜ࡋ࡚㸪㌿⛣ࡀ⏕ࡌࡿࡲ࡛ࡢ཯ᛂ᫬㛫ࢆ ᐃࡋࡓࠋᐇ㦂 1 ࡛ࡣ㸪ᐇ㦂ཧຍ⪅ࡣᏛ⩦ẁ㝵࡛౛㢟ࢆ 2 ࡘゎ࠸ࡓᚋ㸪㌿⛣ࢸࢫࢺࢆཷࡅࡓࠋࡑࡢ⤖ᯝ㸪࢖ࣛࢫࢺࡢ
᭷↓࡜᜛ពᛶࡢ᭷↓ࡢ࠸ࡎࢀࡶṇ⟅ᩘ࡜཯ᛂ᫬㛫࡟᭷ព࡞ᕪࡀㄆࡵࡽࢀ࡞࠿ࡗࡓࠋᐇ㦂 2 ࡛ࡣ㸪ᐇ㦂 1 ࡢᏛ⩦ẁ㝵ࡢ౛㢟ࢆ 1 ࡘῶࡽࡋࡓࡇ࡜௨እ㸪ᡭ⥆ࡁࡣྠࡌ࡛࠶ࡗࡓࠋࡑࡢ⤖ᯝ㸪࢖ࣛࢫࢺࡀ࡞࠸᪉ࡀ࢖ࣛࢫࢺࡀ࠶ࡿሙྜࡼࡾࡶ㸪㌿⛣ࢸࢫࢺࡢṇ⟅ᩘࡀ㧗ࡃ࡞ࡗࡓࡀ㸪཯ᛂ᫬㛫࡟㐪࠸ࡣㄆࡵࡽࢀ࡞࠿ࡗࡓࠋࡇࢀࡽࡢ⤖ᯝࡣ㸪࢖ࣛࢫࢺࡣグྕࡼࡾࡶ᝟ሗ㔞ࡀከ࠸ࡓࡵ㸪⫼ᚋࡢᩘᏛⓗᵓ㐀ࢆㄆ㆑࠶ࡿ࠸ࡣᒎ㛤ࡋ࡙ࡽࡃࡍࡿྍ⬟ᛶࢆ♧ࡋ࡚࠸ࡿࠋ
 Key words: transfer of learning, concrete example, abstract example, arbitrariness௧࿴ඖᖺᗘࡢᏛຊ≧ἣㄪᰝࡢሗ࿌᭩ (ᅜ❧ᩍ⫱ᨻ౛࠼ࡤ㸪୰Ꮫᰯ࣭㧗➼Ꮫᰯ࡟࠾ࡅࡿᩘᏛᩍ⫱ࡢ┠ᶆ⟇◊✲ᡤ, 2019) ࡟࠾࠸࡚㸪୰ᏛᰯᩘᏛࡢศ㔝࡛ࡣ㸪ࡢ 1 ࡘ࡜ࡋ࡚ࠕ஦㇟ࡢᮏ㉁ࡸ௚ࡢ஦㇟࡜ࡢ㛵ಀࢆㄆᇶ♏ⓗ࡞ィ⟬࣭ᴫᛕ⌮ゎ࡟㛵ࡋ࡚ࡣᨵၿࡢഴྥࡀ࠶㆑ࡋ࡚⤫ྜⓗ࣭Ⓨᒎⓗ࡟⪃ᐹࡍࡿຊࠖࢆᥖࡆ࡚࠸ࡿࡿࡀ㸪ᘧኚᙧࡢ┠ⓗ㸪ၥ㢟ゎỴࢆࡍࡿࡓࡵࡢᴫᛕ࣭ἲ(ᩥ㒊⛉Ꮫ┬, 2017, 2018)ࠋ⤫ྜ࣭Ⓨᒎࡢࡓࡵ࡟ࡣ㸪஦๎ࡢ฼⏝࡜ㄝ᫂࡟ㄢ㢟ࡀ࠶ࡿ࡜ᣦ᦬ࡉࢀ࡚࠸ࡿࠋࡲ㇟㛫ࡢ㛵ಀࡸඹ㏻ᛶ࠶ࡿ࠸ࡣᵓ㐀ࢆᢳฟࡍࡿᚲせࡀࡓ㸪Ꮫ⩦ᣦᑟせ㡿 (ᩥ㒊⛉Ꮫ┬, 2017, 2018) ࡛ࡣ㸪୰࠶ࡿࠋࡇࢀࡣ㸪ᢳ㇟໬ (abstraction) ࡜࿧ࡤࢀࡿᛮ⪃㐣Ꮫᰯ࣭㧗➼ᏛᰯࡢᩘᏛᩍ⫱࡟࠾࠸࡚㔊ᡂࡍ࡭ࡁࡶࡢ⛬࡛࠶ࡿࠋゝ࠸᥮࠼ࢀࡤ㸪ᩥ❶㢟࡞࡝ࡢタၥ࡟࠾࠸࡜ࡋ࡚ࠕᩘᏛⓗぢ᪉ࠖ࡜ࠕᩘᏛⓗ⪃࠼᪉ࠖࢆᣲࡆ࡚࠸࡚⾲㠃ⓗ࡞ෆᐜࡀ␗࡞ࡗ࡚࠸࡚ࡶ㸪௨๓࡟ྲྀࡾ⤌ࢇࡿࠋᩘᏛⓗぢ᪉ࡣࠕᴫᛕ➼࡟╔┠ࡋ࡚㸪ࡑࡢ≉ᚩࡸᮏࡔၥ㢟࡜ࡢᵓ㐀ⓗ㢮ఝᛶ࡟Ẽ࡙࠸ࡓࡾ㸪ၥ㢟ࡢタᐃ㉁ࢆᤊ࠼ࡿࡇ࡜ࠖ㸪ᩘᏛⓗ⪃࠼᪉ࡣࠕ᪤⩦ࡢ▱㆑ཬࡧ࠿ࡽ⫼ᚋࡢᵓ㐀ࢆぢᢤ࠸࡚࡝ࡢ▱㆑ࢆ㛵㐃࡙ࡅࡿ࡭ᢏ⬟ࢆ㛵㐃࡙ࡅ⤫ྜⓗ࣭Ⓨᒎⓗ࡟⪃ᐹࡍࡿࡇ࡜ࠖ࡜ࡁ࠿ࢆุ᩿ࡋࡓࡾࡍࡿ㐣⛬࡛࠶ࡿࠋࡇࡢព࿡࡛㸪ᩘᐃ⩏ࡉࢀ࡚࠸ࡿࠋࡇࢀࡽࡢ⏝ㄒࢆ⏝࠸ࡿ࡞ࡽࡤ㸪ᏛᏛᩍ⫱࡟࠾࠸࡚⩦ᚓࡀᮇᚅࡉࢀࡿࡢࡣ㸪ࠕᏛ⩦ࡢ㌿⛣ຊ≧ἣㄪᰝࡢሗ࿌᭩ࡣ㸪ᩘᏛⓗぢ᪉ࡼࡾࡶࡴࡋࢁᩘ

(transfer of learning)࡛ࠖ࠶ࡿ࡜ゝ࠼ࡿࠋᏛ⩦ࡋࡓࡇ࡜Ꮫⓗ⪃࠼᪉ࡢ⩦ᚓࢆ࠸࠿࡟ಁࡍ࠿ࡀ㸪௒ᚋࡢᩘᏛᩍࢆ௚ࡢᏛ⩦࠶ࡿ࠸ࡣ᪂ࡓ࡞ㄢ㢟࡟ຠ⋡ⓗ࡟฼⏝࣭᥼⫱࡟࠾ࡅࡿㄢ㢟࡛࠶ࡿ࡜࠸࠺ࡇ࡜ࡀ࡛ࡁࡿࠋ⏝࡛ࡁࡿࡼ࠺࡟࡞ࡿࡇ࡜ࡀࠕ⤫ྜⓗ࣭Ⓨᒎⓗ࡛ࠖ࠶Ꮫ⩦ᣦᑟせ㡿࡛ࡣ㸪ᩘᏛⓗ⪃࠼᪉࡟㝈ࡽࡎ㸪ࠕ⤫ྜࡾ㸪ࠕ㌿⛣࡛ࠖ࠶ࡿࠋࡋࡓࡀࡗ࡚㸪ᩘᏛⓗ⪃࠼᪉ࡢ⩦ⓗ࣭Ⓨᒎⓗࠖ࡜࠸࠺⾲⌧ࡀࡋࡤࡋࡤ⏝࠸ࡽࢀ࡚࠸ࡿࠋᚓ࡜࠸࠺ᩘᏛᩍ⫱ࡢㄢ㢟ࡣ㸪Ꮫ⩦ࡢ㌿⛣ࢆಁࡍࡓࡵ 

ࡢ᪉ἲㄽ࡟࢔ࣉ࣮ࣟࢳࡋ࡚࠸ࡃᚲせࡀ࠶ࡿࠋࡇࢀ࡟㛵ࢃࡿືྥ࡜ࡋ࡚㏆ᖺ㸪ࠕᩍ⛉᭩࡛ࡢ࢖ࣛࢫ

Table 1
Kaminski et al. (2005) ࡀ⏝࠸ࡓලయ౛࡜ᢳ㇟౛ࢺ฼⏝ࠖ࡬ࡢ㛵ᚰࡀ㧗ࡲࡗ࡚࠸ࡿࠋ㔠⏣ (2012) ࡣ㸪ලయⓗ౛♧᪥ᮏ࡟ẚ࡭࡚ࣇ࢕ࣥࣛࣥࢻࡢᩍ⛉᭩࡛ࡣ࢖ࣛࢫࢺࡢせ⣲㔞ࡀ㇏ᐩ࡛࠿ࡘࡑࡢ✀㢮ࡶከᵝ࡛࠶ࡿ࡜㏙࡭࡚࠸ࡿࠋ

₇⟬つ๎⤖ྜせ⣲ᢳ㇟ⓗ౛♧⤖ᯝ⤖ྜせ⣲⤖ᯝ

ࡇࡢẚ㍑ࡣᑠᏛᰯᩍ⛉᭩࡟␃ࡲࡗ࡚࠸ࡿࡶࡢࡢ㸪ከᵝ࡞࢖ࣛࢫࢺ࡟ࡼࡾ㸪ᚓࡽࢀࡓ᝟ሗࢆ㐺ษ࡟⌮ゎࡋ࡚ၥ㢟ゎỴࢆᅗࡿ⬟ຊࡸ㸪ᩘᏛⓗ࡞▱㆑ࡀᵝࠎ࡞ሙ㠃࡛౑⏝࡛ࡁࡿ࡜࠸࠺ᛂ⏝ྍ⬟ᛶࡢ⌮ゎ࡟ࡘ࡞ࡀࡿὀ㸸ᐇ㦂ࡢタᐃ࡛ࡣ㸪࡜᥎ᐹࡋ࡚࠸ࡿࠋᢅࢃࢀ࡚࠸ࡿࠋ࡜ࡣ༢఩ඖ (identity) ࡜ࡋ࡚ࡲࡓ㸪ᩍ⛉᭩࡟ࡣ࢖ࣛࢫࢺࡔࡅ࡛࡞ࡃ෗┿ࡶ⏝࠸ࡽࢀ࡚࠸ࡿࡀ㸪ᖹᡂ 14 ᖺ࠿ࡽ඲࡚ࡢᩍ⛉᭩ࡀ඲࣮࣌
ࢪ࣮࢝ࣛ༳ๅ࡜࡞ࡗࡓ (ྜྷ⏣, 2016)ࠋࡑࡢࡓࡵ㸪࢖ࣛ(2011) ࡣ㸪Kaminski et al. (2005) ࡢᐇ㦂ᡭ⥆ࡁ࡟ᑐࡋ㸪ࢫࢺࡸࢢࣛࣇ࡞࡝㸪⥺✀࡛ࡣ࡞ࡃⰍ࡛༊ู࡛ࡁࡿࡼձ౛♧ࡀලయⓗ࠿ᢳ㇟ⓗ࠿࡜࠸࠺௨እ࡟࢖ࣛࢫࢺ࡛࠺࡟࡞ࡾ㸪⨾ࡋࡃ౛♧ࡉࢀ㸪ෆᐜࡀ⌮ゎࡋࡸࡍࡃ࡞♧ࡉࢀࡓࡶࡢ࡟ᑐࡍࡿ᧯సᛶࡢศ࠿ࡾࡸࡍࡉࡢ㐪࠸ࡗࡓ (⌧ᅾࡢᩍ⛉᭩ࡣⰍᙅࡢඣ❺࣭⏕ᚐ࡬ࡢ㓄៖࠿ࡀ஺⤡ࡋ࡚࠾ࡾ㸪ࡑࢀ࡟ࡼࡾ㌿⛣࡟࠾࠸࡚୙බᖹࡀ

ࡽ࣮࢝ࣛࣘࢽࣂ࣮ࢧࣝࢹࢨ࢖ࣥ࡟ᇶ࡙࠸࡚ไసࡉࢀ⏕ࡌ࡚࠸ࡿࡇ࡜㸪ղᏛ⩦࡟࠾ࡅࡿᢳ㇟ⓗ౛♧࠿ࡽᐇ࡚࠸ࡿ)ࠋࡇࡢⅬ࡛㸪༳ๅᢏ⾡ࡸࣘࢽࣂ࣮ࢧࣝࢹࢨ࢖㦂ཧຍ⪅ࡣ୍యఱࢆᐇ㝿࡟Ꮫࡧྲྀࡗࡓࡢ࠿࡜࠸࠺ࡇࣥࡣ㸪࢖ࣛࢫࢺࡸ෗┿࡟ࡼࡿලయⓗ౛♧࡟ᢿ㌴ࢆ᥃࡜ࡢ 2 ࡘࡢᢈุⅬࢆᣦ᦬ࡋࡓࠋࡑࡢୖ࡛㸪De Bock etࡅ࡚࠸ࡿ࡜ゝ࠼ࡿࠋal.ࡣ Kaminski et al.࡜ྠᵝࡢලయ౛࣭ᢳ㇟౛ࢆ฼⏝ࡋ㸪ࡇࡢࡼ࠺࡟㸪᪥ᮏࡢᩍ⛉᭩ࡢ࢖ࣛࢫࢺࡀ඘ᐇࡋ࡚㌿⛣ඛࡢၥ㢟࡟ᢳ㇟౛ࡔࡅ࡛࡞ࡃලయ౛ࡶ㏣ຍࡋᐇ࠸ࡃ୍᪉㸪ᩘᏛᩍ⫱࡟㛵ࡍࡿᾏእࡢ◊✲࡛ࡣ࢖ࣛࢫ

㦂ࢆ⾜ࡗࡓ࡜ࡇࢁ㸪Ꮫ⩦ࢸࢫࢺࡣලయⓗ౛♧ࡢ᪉ࡀࢺࡢ฼⏝ࡀᚲࡎࡋࡶⰋ࠸ຠᯝࢆࡶࡓࡽࡍ࡜ࡣ㝈ࡽ࡞ᢳ㇟ⓗ౛♧ࡼࡾࡶࣃࣇ࢛࣮࣐ࣥࢫࡀඃࢀ㸪ࡉࡽ࡟㌿࠸ࡇ࡜ࡀ♧ࡉࢀ࡚ࡁ࡚࠸ࡿࠋ⛣ࢸࢫࢺࡀලయⓗ౛♧࡛࠶ࡿሙྜࡣ㸪Ꮫ⩦ࢸࢫࢺࡀ

Kaminski, Sloutsky, & Heckler (2005) ࡣ㸪⩌ㄽ (groupලయⓗ౛♧࡛࠶ࡗࡓ᪉ࡀᢳ㇟ⓗ౛♧ࡢሙྜࡼࡾࡶࣃtheory) ࡢ๫వ₇⟬ (modulo) ࢆᢅࡗࡓၥ㢟࡟࠾࠸࡚㸪ࣇ࢛࣮࣐ࣥࢫࡀ㧗ࡃ࡞ࡿࡇ࡜ࢆ♧ࡋࡓࠋࡇࡢࡼ࠺࡟Ꮫ⩦ẁ㝵࡛ලయⓗ࡞౛♧ (ィ㔞࢝ࢵࣉ࡟ධࡗ࡚࠸ࡿࢸࢫࢺᚓⅬࡢഃ㠃࡛ぢࢀࡤ㸪ලయⓗ౛♧ࡢ᪉ࡀ᭷฼Ỉ) ࡜ᢳ㇟ⓗ࡞౛♧ (グྕ) ࡢ㐪࠸ཬࡧグྕࡸලయ࡛࠶ࡿࡀ㸪ࡇࢀࡣලయᛶࡀ㧗࠸ࡇ࡜࡟⏤᮶ࡍࡿᢕᥱⓗ࡞౛♧ࡢⰍ࡞࡝ࡢ▱ぬ᝟ሗࡢከࡉࡢ㐪࠸ࡀ㸪Ꮫ⩦ࡢࡋࡸࡍࡉ࡟ࡼࡿࡶࡢ࡜⪃࠼ࡽࢀࡿࠋࡋ࠿ࡋ㸪ᢳ㇟ࡢࡋࡸࡍࡉࡸ㌿⛣ࡢࡋࡸࡍࡉ࡟ᙳ㡪ࡍࡿࡢ࠿᳨ウࡋⓗ౛♧࡛ࡢᏛ⩦ࢸࢫࢺࢆ⤒㦂ࡋࡓ⩌ࡣ㸪ලయⓗ㸭ᢳࡓࠋࡼࡾලయⓗ࡟ࡣ㸪Table 1 ࡟♧ࡍࡼ࠺࡞せ⣲࡜ࡑ㇟ⓗ౛♧࡟ࡼࡿ㌿⛣ࢸࢫࢺ୧᪉࡛ᚓⅬࢆ㧗ࡵࡓࡀ㸪ࢀ࡟ࡘ࠸࡚ࡢ₇⟬つ๎ࡢ౛࠿ࡽ㸪௚ࡢ⤌ࡳྜࢃࡏ࡟ලయⓗ౛♧࡛ࡢᏛ⩦ࢸࢫࢺࢆ⤒㦂ࡋࡓ⩌ࡣ㸪ᢳ㇟ⓗࡘ࠸࡚ኚ᥮ࢆ⾜࠺࡜࡝ࡢࡼ࠺࡞⤖ᯝ࡜࡞ࡿ࠿ࢆุ᩿౛♧࡟ࡼࡿ㌿⛣ࢸࢫࢺ࡛኱ࡁࡃᚓⅬࢆୗࡆࡓࠋࡍࡿㄢ㢟ࢆ⏝࠸ࡓᐇ㦂࡛࠶ࡗࡓࠋࡑࡢ⤖ᯝ㸪ලయⓗࡇࡢ De Bock et al. (2011) ࡢ⤖ᯝࡣ㸪ᢳ㇟ⓗ౛♧࡟౛♧ࡣᢳ㇟ⓗ౛♧ࡼࡾࡶᏛ⩦ࡉࢀࡸࡍ࠸཯㠃㸪Ꮫ⩦ᑐࡍࡿ㌿⛣࡜࠸࠺ഃ㠃࡛ぢࢀࡤ㸪Kaminski et al. (2005)ࡢ㌿⛣ࡢ⏕ࡌࡸࡍࡉࢆጉࡆࡓࠋࡉࡽ࡟㸪▱ぬ᝟ሗࡀࡢ▱ぢ࡜ྠࡌ࡛࠶ࡿࠋࡍ࡞ࢃࡕ㸪ලయ౛ࡼࡾࡶᢳ㇟ᑡ࡞࠸౛♧ࡢ᪉ࡀከ࠸౛♧ࡼࡾࡶ㌿⛣ࡀ⏕ࡌࡸࡍ࠸౛ࡢ᪉ࡀ㸪ᢳ㇟౛࡟ᑐࡋ࡚㌿⛣ࡀ⏕ࡌࡸࡍ࠸࡜࠸࠺ࡇ࡜ࡀ᫂ࡽ࠿࡟࡞ࡗࡓ (ලయ౛࡟ࡼࡗ࡚ᢳ㇟ⓗ࡞⪃ࡇ࡜࡛࠶ࡿࠋࡇࡢⅬࡣ㸪ᩘᏛࡢ୰ᚰⓗᛶ㉁ࡀᢳ㇟ᛶ

࠼᪉ࡢᏛ⩦ࡀጉࡆࡽࢀࡿྍ⬟ᛶࢆᣦ᦬ࡋ࡚࠸ࡿࣞࣅࡸᙧᘧᛶ㸪୍⯡ᛶ࡟࠶ࡿࡇ࡜ (e.g., MacLane, 1986 ㉥࣮ࣗ࡜ࡋ࡚㸪Brown, McNeil, & Glenberg , 2009)ࠋ

ᑿ࣭ᒸᮏヂ 2019) ࢆ㋃ࡲ࠼ࡿ࡜㸪ඛ㏙ࡋࡓᩘᏛᩍ⫱De Bock, Deprez, Van Dooren, Roelens, & Verschaffelࡢ┠ᶆࡢ㐩ᡂࡍࡿࡓࡵࡢ᪉ἲㄽⓗᡭࡀ࠿ࡾ࡜ࡋ࡚㠀

 ᖖ࡟㔜せ࡛࠶ࡿ࡜ᛮࢃࢀࡿࠋࡋ࠿ࡋ࡞ࡀࡽ㸪De Bockᐇ㦂 1et al. ࡀᣦ᦬ࡋ࡚࠸ࡿࡼ࠺࡟㸪ᢳ㇟ⓗ౛♧ࡢ࡝ࡢࡼ࠺

᪉ἲ࡞≉ᚩࡀ㌿⛣ࢆ⏕ࡌࡸࡍࡃࡋ࡚࠸ࡿࡢ࠿ࡣุ↛࡜ࡋᐇ㦂ཧຍ⪅  ᒱ㜧኱Ꮫᩍ⫱Ꮫ㒊⏕ 40 ྡ (⏨ᛶ 14࡞࠸ࠋྡ㸪ዪᛶ 26 ྡ)ࠋᖹᆒᖺ㱋ࡣ 20.73 ṓ (SD = 0.56)ࠋ

Table 1 ࡟♧ࡉࢀ࡚࠸ࡿࡼ࠺࡟㸪ලయ౛࡜ࡋ࡚⏝࠸ᐇ㦂ィ⏬  ౛♧࡛ࡢࠕ࢖ࣛࢫࢺࠖ(2 Ỉ‽㸸࠶ࡾ vs.

ࡽࢀࡓࠕィ㔞࢝ࢵࣉ࡜Ỉࡢ㔞ࠖࡣ㸪De Bock et al. (2011)࡞ࡋ) ཬࡧᚠ⎔ⓗᛶ㉁࡟࠾ࡅࡿࠕ᜛ពᛶࠖ(2 Ỉ‽㸸࠶ࡀᣦ᦬ࡋࡓ㏻ࡾ㸪᧯సᛶࡢศ࠿ࡾࡸࡍࡉࡶࡉࡿࡇ࡜

ࡾ vs. ࡞ࡋ) ࢆ⊂❧ኚᩘ㸪ṇ⟅⋡ཬࡧつ๎ᛶࢆぢᢤ࡞ࡀࡽ㸪᪥ᖖⓗ࡟┠࡟ࡍࡿࣔࣀ࡜㔞ⓗ࡛ᚠ⎔ⓗ࡞ኚࡁゎࡃࡲ࡛ࡢ཯ᛂ᫬㛫ࢆᚑᒓኚᩘ࡜ࡍࡿ 2 せᅉཧຍ໬࡟ᑐᛂࡉࡏࡽࢀࡿࡇ࡜ࡀ┤ឤⓗ࡟ศ࠿ࡿ࡜࠸࠺≉⪅㛫ィ⏬ࠋ௨㝆㸪࢖ࣛࢫࢺ࠶ࡾࢆ I㸪࢖ࣛࢫࢺ࡞ࡋࢆᚩࡀ࠶ࡿࠋࡑࡢព࿡࡛㸪ィ㔞࢝ࢵࣉࡢ⤮ (Ỉࡢ㔞) ࡢNI㸪᜛ពᛶ࠶ࡾࢆ A㸪᜛ពᛶ࡞ࡋࢆ NA ࡜␎グࡍࡿࠋᚠ⎔ᛶ࡟ࡣ modulo ᵓ㐀ࢆ⮬↛࡟ᙜ࡚ࡣࡵࡿࡇ࡜ࡀ4 ࡘࡢ⩌ (I-A ⩌㸪NI-A ⩌㸪I-NA ⩌㸪NI-NA ⩌) ࡟࡛ࡁࡿ࡜ᛮࢃࢀࡿࠋ୍᪉㸪グྕࡢᚠ⎔ࡣᐇ㦂࡟࠾ࡅࡑࢀࡒࢀ 10 ྡࡢᐇ㦂ཧຍ⪅ࢆ↓సⅭ࡟๭ࡾᙜ࡚ࡓࠋࡿேᕤⓗ࡞タᐃ࡟㐣ࡂ࡞࠸ࠋࡑࡢព࿡࡛㸪ᢳ㇟౛࡟ᐇ㦂ㄢ㢟  Kaminski (2017) ࡟‽ᣐࡋ㸪⩌ㄽࡢ๫వࡣࠕ᜛ពᛶ (arbitrariness)ࠖࡀ࠶ࡿࠋࡋ࠿ࡋ࡞ࡀࡽ㸪₇⟬ (modulo) ࢆ㢟ᮦ࡜ࡋࡓࠋ
ྛ⩌ࡢ᧯స࡜ࡋ࡚⏝࠸

Kaminski et al. (2005) ࡸ De Bock et al. (2011) ࢆྵࡵ㸪ࡓ࢖ࣛࢫࢺཬࡧグྕࢆ Table 2 ࡟♧ࡍ (I-NA ⩌࡜ NI-ල య ౛࡜ ᢳ㇟ ౛ࡢ ㌿⛣ ࢆẚ ㍑ ࡍࡿ ◊✲ (e.g.,NA ⩌ࡣ᜛ពᛶࡀ࡞࠸᮲௳࡛࠶ࡾ㸪ಙྕࡢኚ໬࡜᳜

Kaminski, 2017; Kaminski, Sloutsky, & Heckles, 2008) ࡟≀ࡢᡂ㛗㐣⛬࡟⮬↛࡞ᚠ⎔ࡀᡂࡾ❧ࡗ࡚࠸ࡿࡢ࡟ᑐ࠾࠸࡚᜛ពᛶࡣ඲ࡃゝཬࡉࢀ࡚࠸࡞࠸ࠋࡋ㸪I-A ⩌࡜ NI-A ⩌ࡢ࢖ࣛࢫࢺཬࡧグྕ࡟ࡣࡑࡢࡼ

ࡋࡓࡀࡗ࡚㸪ᐇ㦂࡛⏝࠸ࡽࢀࡓලయⓗ౛♧ࡀഛ࠼࠺࡞ᚠ⎔ࡀ᜛ពⓗ࡛࠶ࡿ)ࠋ࡞࠾㸪NI-A ⩌࡜ NI-NA࡚࠸ࡿලయᛶࡣ㸪ձグྕ࡛ࡣ࡞ࡃ࢖ࣛࢫࢺ࡛࠶ࡿ࡜

⩌ࡢグྕࡣ㸪࢖ࣛࢫࢺࢆ⏝࠸ࡿ⩌࡜ᑐᛂࡉࡏࡿࡓࡵ࠸࠺ࠕ⾲㠃ⓗ࡞ഃ㠃ࠖཬࡧղ᪥ᖖⓗ࡞⤒㦂࡟༶ࡋࡓ㯮࡛ࡣ࡞ࡃ࣮࢝ࣛࢆࡘࡅࡓࠋࡇࢀࡽࡢ࢖ࣛࢫࢺཬࡧ

modulo ࡢᚠ⎔ⓗᛶ㉁ࢆႏ㉳ࡉࡏࡿ࡜࠸࠺ࠕ㠀᜛ពⓗグྕࢆ⏝࠸࡚㸪౛㢟ࣉࣜࣥࢺ 2 ᯛཬࡧ㌿⛣ࢸࢫࢺ 1࡞ഃ㠃ࠖ࡜࠸࠺ 2 ࡘࡢഃ㠃ࡀศ㞳ࡉࢀ࡚࠸࡞࠸࡜⪃ᯛࢆసᡂࡋࡓࠋ1 ᯛ┠ࡢ౛㢟ࣉࣜࣥࢺࡣ㸪3 ✀ࡢ࢖ࣛ࠼ࡽࢀࡿࠋࡘࡲࡾ㸪ලయⓗ౛♧ࡣࠕᚠ⎔ⓗᛶ㉁ࡀ㠀᜛ࢫࢺࡲࡓࡣグྕ࡟ࡘ࠸࡚㸪ձ඲࡚ࡢ⤌ࡳྜࢃࡏ࡛ࡢពⓗ࡞࢖ࣛࢫࢺ࡛ࠖ࠶ࡾ㸪㏫࡟㸪ᢳ㇟ⓗ౛♧ࡣࠕᚠ⎔

₇⟬つ๎㸪ղ㛵ಀᛶࢆ♧ࡋࡓᚠ⎔ᅗ㸪ճmodulo 3 ࡢⓗᛶ㉁ࡀ᜛ពⓗ࡞グྕࠖ࡜ࡳ࡞ࡏࡿࡔࢁ࠺ࠋࡋࡓࡀ

0㸪1㸪2 ࡬ࡢᑐᛂ࡙ࡅ࡜ᚠ⎔ⓗᛶ㉁࡟㛵ࡍࡿ࠸ࡃࡘ࠿ࡗ࡚㸪⾲㠃ⓗ࡞ഃ㠃࡜᜛ពⓗ࡞ഃ㠃ࢆศࡅࡿ࡞ࡽࡤ㸪ࡢ౛㸪մ₇⟬ࡢ⦎⩦ၥ㢟 3 㢟࠿ࡽᵓᡂࡉࢀࡓࠋ2 ᯛ┠ࠕᚠ⎔ⓗᛶ㉁ࡀ᜛ពⓗ࡞࢖ࣛࢫࢺࠖ࡜ࠕᚠ⎔ⓗᛶ㉁ࡢ౛㢟ࣉࣜࣥࢺࡣ㸪ձ3 ✀ࡢ࢖ࣛࢫࢺࡲࡓࡣグྕࡢࡀ㠀᜛ពⓗ࡞グྕࠖ࡜࠸࠺せ⣲ࡶ᝿ᐃࡍࡿࡇ࡜ࡀ࡛㛵ಀᛶࢆぢࡘࡅࡿࡓࡵ࡟᭱ప㝈ᚲせ࡜࡞ࡿ 4 ࡘࡢ₇ࡁࡿࡔࢁ࠺ࠋࡑࡇ࡛ᮏ◊✲ࡣ㸪ලయᛶࢆᣢࡘ࢖ࣛࢫࢺࡢ᭷↓࡜᜛ពᛶࡢ᭷↓ࡢ㐪࠸࡟ࡼࡿ㌿⛣ࡢ⏕ࡌࡸࡍࡉࢆ᳨ウ

Table 2ࡍࡿࡇ࡜ࢆ┠ⓗ࡜ࡍࡿࠋྛ⩌ࡢ౛㢟ࣉࣜࣥࢺ࡟⏝࠸ࡓ࢖ࣛࢫࢺཬࡧグྕ ௬ㄝࡣ௨ୗࡢ㏻ࡾ࡛࠶ࡿࠋ➨ 1 ࡟㸪グྕࢆ౛࡟Ꮫ

⩌⩦ࡍࡿ᪉ࡀ࢖ࣛࢫࢺࢆ౛࡟Ꮫ⩦ࡍࡿࡼࡾࡶ㸪ᢳ㇟౛

1 ᯛ┠

I-ANI-AI-NANI-NA࡟ࡼࡿ㌿⛣ࢸࢫࢺ࡟࠾࠸࡚ṇ⟅ᩘࡀከࡃ㸪཯ᛂ᫬㛫ࡀ▷ࡃ࡞ࡿ࡛࠶ࢁ࠺ࠋ➨ 2 ࡟㸪ᚠ⎔ⓗᛶ㉁ࡀ᜛ពⓗ࡛࠶ࡿ౛ࢆᏛ⩦ࡍࡿ᪉ࡀ᜛ពⓗ࡛࡞࠸౛ࢆᏛ⩦ࡍࡿ2 ᯛ┠ࡼࡾࡶ㸪ᢳ㇟౛࡟ࡼࡿ㌿⛣ࢸࢫࢺ࡟࠾࠸࡚ṇ⟅ᩘࡀከࡃ㸪཯ᛂ᫬㛫ࡀ▷ࡃ࡞ࡿ࡛࠶ࢁ࠺ࠋ
ὀ㸸ࡑࢀࡒࢀ᭱ึࡢ࢖ࣛࢫࢺ㸭グྕࢆ༢఩ඖ࡜ࡋ࡚ᢅࡗࡓࠋ 




Figure 1. ౛㢟ࣉࣜࣥࢺࡢ౛ (࢖ࣛࢫࢺ࠶ࡾ࠿ࡘ᜛ពᛶ࠶ࡾࡢ⩌ࡢሙྜ)ࠋᕥࡀ 1 ᯛ┠㸪ྑࡀ 2 ᯛ┠ࠋ௚ࡢ⩌࡟ࡘ࠸࡚ࡶྠᵝ࡟㸪Table 2 ࡢ࢖ࣛࢫࢺࡲࡓࡣグྕࢆ⏝࠸࡚౛㢟ࣉࣜࣥࢺࢆసᡂࡋࡓࠋ⟬つ๎㸪ղ₇⟬ࡢ⦎⩦ၥ㢟 3 㢟࠿ࡽᵓᡂࡉࢀࡓࠋ౛

ᐇ㦂ཧຍ⪅࡟㸪ࣃࢬࣝࡣ 3 ✀ࡢ࢖ࣛࢫࢺࡲࡓࡣグ㢟ࣉࣜࣥࢺࡢ୍౛࡜ࡋ࡚㸪࢖ࣛࢫࢺ࠶ࡾ࠿ࡘ᜛ពᛶ

ྕ࠿ࡽ࡛ࡁ࡚࠸ࡿࡇ࡜㸪9 ࡘࡢ₇⟬つ๎ࡀ࠶ࡿࡇ࡜࠶ࡾࡢ⩌࡟౑⏝ࡋࡓࡶࡢࢆ Figure 1 ࡟♧ࡍࠋࢆ౛㢟ࣉࣜࣥࢺࡢヱᙜ⟠ᡤࢆᣦ࡛♧ࡋ࡞ࡀࡽᩍ♧ࡋ඲࡚ࡢ⩌࡛ඹ㏻ࡢ㌿⛣ࢸࢫࢺࡣ㸪Kaminski (2017)ࡓࠋ3 ✀ࡢ࢖ࣛࢫࢺࡲࡓࡣグྕ࡟ᩘᏐࢆ๭ࡾᙜ࡚ࡿ

࡜ྠᵝ࡟㸪4 ✀ࡢグྕ‫۔‬㸪ѭ㸪‫ۼ‬㸪‫ࡢۍ‬㛵ಀᛶࢆぢࡘࡇ࡜࡛⫼ᚋࡢᚠ⎔ᛶࢆ☜ㄆࡋ (Figure 1 ཧ↷)㸪₇⟬つࡅࡿࡓࡵ࡟᭱ప㝈ᚲせ࡜࡞ࡿ 5 ࡘࡢ₇⟬つ๎࡜タၥ๎ࡀᡂࡾ❧ࡘࡇ࡜ࢆㄝ᫂ࡋࡓࠋ㉁ၥࡀ࠶ࡿ࠿ࢆ☜ㄆ

10 㢟࠿ࡽᵓᡂࡉࢀࡓ (Table 3)ࠋタၥ (1) 㹼 (3) ࡣ₇ᚋ㸪ᐇ㦂ཧຍ⪅ࡣ⦎⩦ࡢࡓࡵ࡟ 3 ၥ࡟ྲྀࡾ⤌ࢇࡔࠋ⟬つ๎ࡢࡳ࡛ゎࡃࡇ࡜ࡀ࡛ࡁࡿၥ㢟㸪タၥ (4) ࡣ₇

ゎࡁ⤊ࢃࡿ࠿㸪࡝࠺ࡋ࡚ࡶゎࡅ࡞࠸࡜࠸࠺᫬Ⅼ࡛㸪⟬つ๎࠿ࡽ modulo ᵓ㐀ࢆぢฟࡍࡇ࡜࡛ึࡵ࡚ゎࡃࡇ࡜ࡀ࡛ࡁࡿၥ㢟㸪タၥ (5) 㹼 (7) ࡣ㛗ࡉࡀ 4 ࡢグྕิࡢၥ㢟㸪タၥࡣ (8) ཬࡧ (9) ࡣ㛗ࡉࡀ 5 ࡢグྕ

Table 3ิࡢၥ㢟㸪タၥ (10) ࡣ㛗ࡉࡀ 6 ࡢグྕิࡢၥ㢟࡛࠶㌿⛣ࢸࢫࢺ࡟࠾ࡅࡿ₇⟬つ๎࡜タၥࡗࡓࠋ₇⟬つ๎ᡭ⥆ࡁ  ึࡵ࡟㸪ᐇ㦂ཧຍ⪅࡟ཧຍྠព᭩ࢆ㓄ᕸࡋ㸪ᛶู㸪ᖺ㱋ࢆྵࡴᐇ㦂ࢹ࣮ࢱࡣ⤫ィⓗ࡟ࡢࡳ฼‫ۍۍ‬Ѝ‫ۼۼ  ۼ‬Ѝѭ  ‫ۍ‬ѭЍ‫ۍ‬‫ۼ‬ѭЍ‫ۼۍ  ۼ‬Ѝ‫۔‬(1) ‫ۍ‬ѭ‫ۍ‬(2) ‫ۼۍ‬ѭ⏝ࡉࢀࡿࡇ࡜㸪㌿⛣ࢸࢫࢺࡢᐇ᪋ࡢ㝿࡟ᡭඖࡢ᧜ᙳ

(3) ‫ۼۼۼ‬(4) ‫ۍ۔۔‬ࢆ⾜࠺ࡇ࡜ࢆㄝ᫂ࡋ㸪ᐇ㝿࡟᧜ᙳࡉࢀࡿ㝿ࡢ᧜ᙳ⏝(5) ‫۔۔ۼ۔‬(6) ѭ‫ۍۍۼ‬ࢫ࣐࣮ࢺࣇ࢛ࣥࡢ࣓࢝ࣛ࢔ࣥࢢࣝࢆ☜ㄆࡉࡏࡓࠋ⨫(7) ‫ۼۼۍ۔‬(8) ‫ۍۼۼ۔ۼ‬ྡ☜ㄆᚋ㸪ᐇ㦂ྠព᭩ࢆᅇ཰ࡋ㸪1 ᯛ┠ࡢ౛㢟ࣉࣜࣥ(9) ‫ۍۍۍ‬ѭѭ(10) ‫۔‬ѭ‫ۍ۔ۼۍ‬ࢺࢆ㓄ᕸࡋࡓࠋタၥὀ㸸ѭࢆ༢఩ඖ࡜ࡋ࡚ᢅࡗࡓࠋ



 

⟅࠼ྜࢃࡏࢆ⾜ࡗࡓࠋᮍᅇ⟅ࡲࡓࡣㄗ⟅ࡀ࠶ࡿሙྜฟࡉ࡞࠸࡜ゎࡃࡇ࡜ࡀ࡛ࡁ࡞࠸࠿ࡽ࡛࠶ࡿࠋࡲࡓ㸪ࡣつ๎ᛶࡢ☜ㄆࢆ⾜࠸㸪㉁ၥࡀ࠶ࡿ࠿ࢆ☜ㄆࡋࡓࠋ཯ᛂ᫬㛫ࡢศᯒࡣ㸪㌿⛣ࢸࢫࢺࡢタၥ (4) ࢆゎࡅ࡞ḟ࡟㸪1 ᯛ┠ࡢ౛㢟ࣉࣜࣥࢺࢆᅇ཰ࡋ㸪2 ᯛ┠ࡢ౛࠿ࡗࡓ I-NA ⩌ࡢ 1 ྡࡢࢹ࣮ࢱࢆ㝖እࡋ࡚⾜ࡗࡓࠋ㢟ࣉࣜࣥࢺࢆ㓄ᕸࡋࡓࠋ2 ᯛ┠ࡢၥ㢟ࡀ 1 ᯛ┠࡜ྠᖹᆒṇ⟅ᩘ࡟ࡘ࠸࡚㸪ཧຍ⪅㛫 2 せᅉศᩓศᯒࢆᵝ࡟ゎࡃࡇ࡜ࡀ࡛ࡁࡿࡇ࡜ࢆᐇ㦂ཧຍ⪅࡟ఏ࠼ࡓࠋ⾜ࡗࡓ⤖ᯝ㸪࢖ࣛࢫࢺࡢ୺ຠᯝ (F (1, 36) = 0.22,ᐇ㦂ཧຍ⪅ࡣ㸪5 ศ㛫㸪2 ᯛ┠ࡢ౛㢟ࣉࣜࣥࢺࡢၥ㢟p = .64)㸪᜛ពᛶࡢ୺ຠᯝ (F (1, 36) = 0.22, p = .64)㸪࢖࡟ྲྀࡾ⤌ࢇࡔࠋ⟅࠼ྜࢃࡏࢆ⾜࠸㸪ᮍᅇ⟅ࡲࡓࡣㄗࣛࢫࢺ࡜᜛ពᛶࡢ஺஫స⏝ (F (1, 36) = 0.08, p = .78)

⟅ࡀ࠶ࡿሙྜ㸪つ๎ᛶࡢ☜ㄆࢆ⾜࠸㸪㉁ၥࡀ࠶ࡿ࠿

ࡢ࠸ࡎࢀࡶ᭷ព࡛ࡣ࡞࠿ࡗࡓࠋ
ᖹᆒ཯ᛂ᫬㛫࡟ࡘ࠸࡚ࡶྠᵝࡢศᯒࢆ⾜ࡗࡓࡀ㸪ࢆ☜ㄆࡋࡓࠋ
2 ᯛ┠ࡢ౛㢟ࣉࣜࣥࢺࢆᅇ཰ࡋࡓᚋ㸪㌿⛣ࢸࢫࢺ࢖ࣛࢫࢺࡢ୺ຠᯝ (F (1, 35) = 0.12, p = .74)㸪᜛ពᛶࡢࢆ㓄ᕸࡋࡓࠋᐇ㦂ཧຍ⪅࡟ࡣ㸪㌿⛣ࢸࢫࢺࡀ౛㢟ࣉ୺ຠᯝ (F (1, 35) = 0.00, p = .99)㸪࢖ࣛࢫࢺ࡜᜛ពᛶࡢࣜࣥࢺ࡜ྠᵝ࡟ゎࡅࡿࡇ࡜㸪౛㢟ࣉࣜࣥࢺ࡜ࡣ␗࡞

஺஫స⏝ (F (1, 35) = 0.05, p = .82) ࡢ࠸ࡎࢀࡶ᭷ព࡛

ࡾ 4 ✀ࡢグྕ࠿ࡽ࡞ࡿࣃࢬ࡛ࣝ࠶ࡿࡇ࡜࡟ὀពࡍࡿ

ࡣ࡞࠿ࡗࡓࠋᖹᆒṇ⟅ᩘࡣ࡝ࡢ⩌ࡶ 10 㢟୰ 9 ၥࢆ㉸ࡇ࡜ࢆᩍ♧ࡋࡓࠋࡉࡽ࡟㸪タၥࡣ 10 ಶ࠶ࡾ㸪᭱኱࡛࠼࡚࠾ࡾ㸪ኳ஭ຠᯝ࡟࡞ࡗ࡚࠸ࡿ࡜⪃࠼ࡽࢀࡿࠋࡇ

10 ศࡢ᫬㛫ࢆ࡜ࡿࡇ࡜㸪ణࡋ㸪10 ศࡼࡾ᪩ࡃ⤊ࢃࡗࡢࡇ࡜ࡣ㸪㌿⛣ࢸࢫࢺࡀ࡝ࡢ⩌ࡢᐇ㦂ཧຍ⪅࡟࡜ࡗࡓሙྜ㸪ࡑࡢ᫬Ⅼ࡛⤊஢࡜ࡍࡿࡇ࡜ࢆㄝ᫂ࡋࡓࠋࡑ࡚ࡶ⥲ࡌ࡚ᐜ࡛᫆࠶ࡗࡓྍ⬟ᛶࢆ♧၀ࡋ࡚࠸ࡿࠋ኱ࡢᚋ㸪ᐇ㦂ཧຍ⪅ࡢᡭඖࡢ᧜ᙳࢆ㛤ጞࡋࡓࠋࢫ࣐࣮Ꮫ⏕ࢆᑐ㇟࡜ࡋࡓ Kaminski (2017) ࡢᐇ㦂ྠᵝ㸪ᮏᐇࢺࣇ࢛ࣥࡢ㘓⏬ࡀṇᖖ࡟⾜ࢃࢀ࡚࠸ࡿࡇ࡜ࢆ☜ㄆࡋ㸪㦂࡛ࡣ㸪Ꮫ⩦ẁ㝵ࡣ modulo 3㸪㌿⛣ࢸࢫࢺࡣ moduloᐇ㦂ཧຍ⪅࡟ 10 ศ࡟タᐃࡉࢀࡓࢱ࢖࣐࣮ࢆぢࡏ࡞

4 ࡛࠶ࡗࡓࠋࡲࡓ㸪Kaminski ࡢᐇ㦂࡟࠾ࡅࡿ㌿⛣ࢸࡀࡽ㸪ࠕࡼ࣮࠸㸪ጞࡵࠖࡢ࠿ࡅኌ࡜࡜ࡶ࡟ゎ⟅ࢆጞࡵࢫࢺࡢᖹᆒṇ⟅⋡ࡣ 41.1㸣࡛࠶ࡗࡓࠋᮏᐇ㦂ࡢ㌿⛣ࡓࠋᐇ㦂⪅ࡣࢱ࢖࣐࣮ࢆᣢࡗ࡚ᐇ㦂ཧຍ⪅࠿ࡽぢ࠼

ࢸࢫࢺ࡜Kaminski ࡢᐇ㦂࡟࠾ࡅࡿ㌿⛣ࢸࢫࢺࡣ࡯ࡰ

࡞࠸఩⨨࡬⛣ືࡋࡓࠋ

ྠ㉁࡛࠶ࡿ࡜⪃࠼ࡽࢀࡿࡓࡵ㸪ᮏᐇ㦂࡟࠾ࡅࡿኳ஭㌿⛣ࢸࢫࢺ⤊஢ᚋ㸪㌿⛣ࢸࢫࢺࢆᅇ཰ࡋ㸪ࢹ࢕ࣈ࣮ࣜࣇ࢕ࣥࢢࢆ⾜࠸㸪ᐇ㦂ෆᐜࢆ௚ࡢே࡟ఏ࠼࡞࠸ຠᯝⓗ࡞ᡂ⦼ࡣ㸪ၥ㢟ࡑࢀ⮬యࡢᐜ᫆ࡉ࡟ࡣᖐᒓ࡛ࡁ࡞࠸࡜⪃࠼ࡽࢀࡿࠋᮏᐇ㦂ࡢཧຍ⪅࡜ Kaminski (2017) ࡢᐇ㦂ࡢཧຍࡼ࠺࡟࠾㢪࠸ࡋ㸪ᐇ㦂ࢆ⤊஢ࡋࡓࠋ⪅ࡢᩘᏛⓗ⬟ຊ࡟㐪࠸ࡀ࡞࠸ࡇ࡜ࢆ௬ᐃࡍࢀࡤ㸪ᮏ⤖ᯝ࡜⪃ᐹ⤖ᯝࡣ㸪౛㢟ࣉࣜࣥࢺ࡟ࡼࡿ⦎⩦ࢆ 2 ẁ㝵࡟ࡋࡓࡇ⩌ࡈ࡜ࡢᖹᆒṇ⟅ᩘ࡜ᖹᆒ཯ᛂ᫬㛫ࢆTable 4 ࡟♧࡜࡛㸪modulo ᵓ㐀࡟ᚲせ࡞ᛮ⪃ࣉࣟࢭࢫ࡟ᑐࡍࡿ័

ࡍࠋ࡞࠾㸪཯ᛂ᫬㛫ࡣ㸪ᐇ㦂⪅࡜௚ 2 ྡࡢィ 3 ྡࡀ㸪ࢀࡀ⏕ࡌࡓ࡜⪃࠼ࡽࢀࡿࠋKaminski ࡢᐇ㦂࡛ࡣゝࢃྛᐇ㦂ཧຍ⪅ࡢゎ⟅ᫎീࢆぢ࡚㸪㛤ጞ࠿ࡽタၥ (4)ࡤ 1 ẁ㝵ࡢ⦎⩦ᡭ⥆ࡁ࡛࠶ࡗࡓࠋᮏᐇ㦂࡛ࡣ౛㢟ࣉࡢᅇ⟅ࢆグධࡋ⤊࠼ࡿࡲ࡛ࡢ᫬㛫࡜ࡋ࡚ィ ࡋࡓࠋࣜࣥࢺࡀ 2 ᯛ㸪ࡍ࡞ࢃࡕᏛ⩦ࢆ 2 ẁ㝵⾜ࡗࡓࡇ࡜࡟

ࡇࢀࡣ㸪タၥ (1) 㹼 (3) ࡜ࡣ␗࡞ࡾ㸪タၥ (4) ࠿ࡽࡼࡾ㸪࡝ࡢ⩌ࡶ㌿⛣ࢸࢫࢺ࡛ࡢᚓⅬࡀୖࡀࡾ㸪⩌㛫ࡣᐇ㦂ཧຍ⪅⮬㌟ࡀ₇⟬つ๎࠿ࡽ modulo ᵓ㐀ࢆぢ࡟ᮏ᮶࠶ࡿ࡜᥎ᐃࡉࢀࡿᡂ⦼ࡢ㐪࠸ࡀ᳨ฟ࡛ࡁ࡞ࡃ࡞ࡗࡓྍ⬟ᛶࡀ࠶ࡿࠋ௨ୖࡢ⪃ᐹ࠿ࡽ㸪ᐇ㦂 2 ࡛ࡣ㸪
2 ᯛ┠ࡢ౛㢟ࣉࣜࣥࢺࢆ㝖ࡁ㸪౛㢟ࣉࣜࣥࢺ 1 ᯛࢆᏛ⩦ࡋࡓᚋ࡟㸪㌿⛣ࢸࢫࢺࢆ⾜࠺ࡼ࠺࡟ᡭ⥆ࡁࢆኚ

Table 4᭦ࡋࡓࠋᐇ㦂 1 ࡟࠾ࡅࡿᖹᆒṇ⟅⋡࡜ᖹᆒ཯ᛂ᫬㛫⩌I-ANI-AI-NANI-NAṇ⟅ᩘ (ၥ)

9.1

9.2

9.2

9.6

ᐇ㦂 2཯ᛂ᫬㛫 (⛊)

214.7

210.3

224.1

202.5᪉ἲᐇ㦂ཧຍ⪅  ᒱ㜧኱Ꮫᩍ⫱Ꮫ㒊 46 ྡ (⏨ᛶ 11 ྡ㸪 ዪᛶ 35 ྡ)ࠋᖹᆒᖺ㱋ࡣ 20.52 ṓ (SD = 0.54)ࠋ

ሗࡀᑡ࡞࠸᪉ࡀ㌿⛣ࡀ⏕ࡌࡸࡍ࠸࡜࠸࠺ Kaminski etᐇ㦂ィ⏬  せᅉィ⏬ࡣᐇ㦂 1 ࡜ྠᵝ࡛࠶ࡗࡓࠋᐇ

al. (2005) ࡜ྠ➼ࡢ⤖ᯝ࡜࡞ࡗࡓࠋ㦂ཧຍ⪅ࢆ↓సⅭ࡟๭ࡾᙜ࡚ࡓ⤖ᯝ㸪I-A ⩌㸪I-NA ⩌ ࡲࡓ㸪᜛ពᛶࡢ୺ຠᯝࡣ㸪ṇ⟅ᩘ࡛ࡶ཯ᛂ᫬㛫࡛

ࡣ 12 ྡ㸪NI-A ⩌㸪NI-NA ⩌ࡣ 11 ྡ࡜࡞ࡗࡓࠋࡶㄆࡵࡽࢀ࡞࠿ࡗࡓࠋ᜛ពⓗ࡛࡞࠸౛♧࡜ࡋ࡚⏝࠸ᐇ㦂ㄢ㢟  ᐇ㦂 1 ࡢ౛㢟ࣉࣜࣥࢺ 1 ᯛ┠࡜㌿⛣ࢸࡓಙྕ࡜ࡑࡢ₎Ꮠ࡟ᑐࡋ࡚㸪ᐇ㦂ཧຍ⪅ࡣࡑࡢᚠ⎔ࢫࢺࢆ⏝࠸࡚⾜ࡗࡓࠋᛶࢆࡍࡄ࡟Ẽ࡙࠸ࡓ࡜ࡋ࡚ࡶ㸪๫వ₇⟬࡜ࡋ࡚ࡢᡭ⥆ࡁ  ᐇ㦂 1 ࡢ 2 ᯛ┠ࡢ౛㢟ࣉࣜࣥࢺ࡟㛵ࡍࡿmodulo ᵓ㐀ࢆᢕᥱࡋ࡟ࡃ࠿ࡗࡓࡓࡵ࠿ࡶࡋࢀ࡞࠸ࠋᩍ♧ࢆ┬࠸ࡓ௨እࡣྠࡌ࡛࠶ࡗࡓࠋ

Kaminski et al. (2005) ࡀලయ౛࡜ࡋ࡚⏝࠸ࡓィ㔞࢝ࢵࣉ (࡜Ỉࡢ㔞) ࡣỈࡢ㔞ࡀ 0㸪1㸪2 ࡜ᩘᏐ࡟⨨ࡁ᥮࠼⤖ᯝ࡜⪃ᐹࡸࡍࡃ modulo ࡢつ๎ᛶ࡟Ẽ࡙ࡁࡸࡍ࠿ࡗࡓ࡜ᛮࢃ⩌ࡈ࡜ࡢᖹᆒṇ⟅ᩘ࡜ᖹᆒ཯ᛂ᫬㛫ࢆTable 5 ࡟♧ࢀࡿࠋᮏ◊✲࡛⏝࠸ࡓಙྕ࡜ࡑࡢ₎Ꮠࡣ㸪ᚠ⎔ࢆࡋࡍࠋ཯ᛂ᫬㛫ࡣ㸪ᐇ㦂 1 ࡜ྠᵝ࡟ࡋ࡚ィ ࡋࡓࠋࡲ࡚࠸ࡿࡶࡢࡢ㸪࡝ࢀࢆ༢఩ඖ࡜ࡋ࡚ࡳ࡞ࡍ࡭ࡁ࠿㸪ࡓ㸪཯ᛂ᫬㛫ࡢศᯒࡣ㸪㌿⛣ࢸࢫࢺࡢタၥ (4) ࢆゎ࠶ࡿ࠸ࡣ࡝ࡢࡼ࠺࡟ᩘᏐ࡟⨨ࡁ᥮࠼ࡿ࡭ࡁ࠿ࡀᐜ᫆

ࡅ࡞࠿ࡗࡓ I-A ⩌ࡢ 2 ྡ㸪NI-A ⩌ࡢ 1 ྡ㸪I-NA ⩌ࡢ࡛࡞࠿ࡗࡓࡓࡵ࡟㸪᜛ពⓗ࡛࡞࠸౛♧࡛࠶ࡗ࡚ࡶ

5 ྡ㸪NI-NA ⩌ࡢ 2 ྡࡢࢹ࣮ࢱࢆ㝖እࡋ࡚⾜ࡗࡓࠋ

moduloࡢつ๎ᛶࢆぢࡘࡅࡿࣉࣟࢭࢫ⮬యࡀ᜛ពⓗ࡞ᖹᆒṇ⟅ᩘ࡟ࡘ࠸࡚㸪ࠕ࢖ࣛࢫࢺࠖࠕ᜛ពᛶࠖࡢ

౛♧࡜࠶ࡲࡾ㐪࠸ࡀ࡞࠿ࡗࡓྍ⬟ᛶࡀ࠶ࡿࠋཧຍ⪅㛫 2 せᅉศᩓศᯒࢆ⾜ࡗࡓ⤖ᯝ㸪࢖ࣛࢫࢺࡢ཯ᛂ᫬㛫࡟࠾࠸࡚ࡣ࠸ࡎࢀࡢ୺ຠᯝ࣭஺஫స⏝ࡶ୺ຠᯝࡀ᭷ព࡛࠶ࡗࡓ (F (1, 42) = 5.20, p = .03,ㄆࡵࡽࢀࡎ㸪཯ᛂ᫬㛫࡟㛵ࡍࡿ௬ㄝࡣᨭᣢࡉࢀ࡞࠿

2

η = 0.11)ࠋ᜛ពᛶࡢ୺ຠᯝ (F (1, 42) = 0.13, p = .73)㸪

ࡗࡓࠋࡋ࠿ࡋ㸪᜛ពⓗ࡛࡞࠸౛♧ࡢ⩌ (I-NA ⩌࡜ NI-࢖ࣛࢫࢺ᜛ពᛶࡢ┦஫స⏝ (F (1, 42) = 0.00, p = .99)

NA ⩌) ࡢ཯ᛂ᫬㛫ࡣ㸪グ㏙⤫ィࡢୖ࡛ࡣ㸪௚ࡢ⩌ࡼࡣ᭷ព࡛ࡣ࡞࠿ࡗࡓࠋࡾࡶᩘ༑⛊ࡣ᪩࠸ࡼ࠺࡟ぢ࠼ࡿࠋࡋ࠿ࡋ㸪タၥ (4) ࡲᖹᆒ཯ᛂ᫬㛫࡟ࡘ࠸࡚ࡶྠᵝࡢศᯒࢆ⾜ࡗࡓࠋ࢖࡛ࡢ཯ᛂ᫬㛫࡜㌿⛣ࢸࢫࢺ඲యࡢṇ⟅ᩘࡣ࡯࡜ࢇ࡝

ࣛࢫࢺࡢ୺ຠᯝ (F (1, 32) = 0.37, p = .55)㸪᜛ពᛶࡢ୺㛵ಀࡀ࡞࠸࡜⪃࠼ࡽࢀ㸪modulo ᵓ㐀ࡢᢕᥱ࡜ά⏝࡟ຠᯝ (F (1, 32) = 2.71, p = .11)㸪࢖ࣛࢫࢺ࡜᜛ពᛶࡢ஺኱ࡁࡃ㛵ࢃࡿࡢࡣ㸪Kaminski (2017; Kaminski et al.,஫స⏝ (F (1, 32) = 0.43, p = .52) ࡢ࠸ࡎࢀࡶ᭷ព࡛ࡣ2005) ࡢ▱ぢྠᵝ㸪ᢳ㇟ⓗ౛♧࡟ࡼࡿᏛ⩦⤒㦂࡛࠶࡞࠿ࡗࡓࠋࡿ࡜࠸࠺ࡇ࡜ࡀ࡛ࡁࡿࡔࢁ࠺ࠋᐇ㦂 2 ࡢ┠ⓗࡣ㸪ᐇ㦂 1 ྠᵝ࡟࢖ࣛࢫࢺࡢ᭷↓࡜᜛ពᛶࡢ᭷↓ࡢ㐪࠸࡟ࡼࡗ࡚㸪ᢳ㇟౛࡬ࡢ㌿⛣ࡢ⏕⥲ྜ⪃ᐹࡌࡸࡍࡉ࡟࡝ࡢࡼ࠺࡞㐪࠸ࡀ࠶ࡿ࠿᳨ウࡍࡿࡇ࡜࡛ᮏ◊✲ࡢ┠ⓗࡣ㸪ලయᛶࢆᣢࡘ࢖ࣛࢫࢺࡢ᭷↓࡜࠶ࡗࡓࠋᐇ㦂ࡢ⤖ᯝ㸪࢖ࣛࢫࢺ࡞ࡋࡢṇ⟅ᩘࡀ࢖ࣛ᜛ពᛶࡢ᭷↓ࡢ㐪࠸࡟ࡼࡿ㌿⛣ࡢ⏕ࡌࡸࡍࡉࢆ᳨ウࢫࢺ࠶ࡾࡢṇ⟅ᩘࡼࡾࡶ㧗ࡃ࡞ࡿࡇ࡜ࡀ♧ࡉࢀࡓࠋࡍࡿࡇ࡜࡛࠶ࡗࡓࠋࡇࢀࡣ㸪ᢳ㇟౛࡟ࡼࡿᏛ⩦ࡢ᪉ࡀලయ౛࡟ࡼࡿᏛ⩦ ᮏ◊✲࡛ࡣ㸪ලయ౛࡛ࡣ࡞ࡃᢳ㇟౛ࢆ⏝࠸ࡿ᪉ࡀ㸪ࡼࡾࡶᢳ㇟ⓗ㌿⛣ࢸࢫࢺ࡛ࡢṇ⟅ᩘࡀ㧗ࡃ㸪どぬ᝟ᢳ㇟ⓗ࡞㌿⛣ࢸࢫࢺ࡟࠾ࡅࡿṇ⟅ᩘࡀ㧗ࡃ࡞ࡗࡓࠋࡇࢀࡣ㸪ලయ౛ࡼࡾࡶ᝟ሗ㔞ࡀᑡ࡞࠸ศ㸪ᢳ㇟౛ࡣ⫼ᚋࡢἲ๎ࡸᩘᏛⓗᵓ㐀࡟ὀពࢆྥࡅࡸࡍࡃࡍࡿ࡞

Table 5ࡾ㸪㌿⛣ࡀ⏕ࡌࡸࡍࡃ࡞ࡗࡓࡓࡵ࡜⪃࠼ࡽࢀࡿᐇ㦂 2 ࡟࠾ࡅࡿᖹᆒṇ⟅ᩘ࡜ᖹᆒ཯ᛂ᫬㛫(Kaminski, 2017; Kaminski et al., 2005)ࠋࡲࡓ㸪De Bock⩌

I-ANI-AI-NANI-NA

et al. (2011) ࡀ♧ࡋࡓ㑅ᢥⓗ࡞㌿⛣⌧㇟㸪ࡍ࡞ࢃࡕ㸪ṇ⟅ᩘ (ၥ)

5.9

8.2

5.6

7.8

ලయ౛࡟ࡼࡿᏛ⩦ࡀᢳ㇟౛࡟ᑐࡋ࡚㈇ࡢ㌿⛣ࢆࡶࡓ཯ᛂ᫬㛫 (⛊)

313.5

311.6

208.1

266.1

ࡽࡍ୍᪉㸪ᢳ㇟౛࡟ࡼࡿᏛ⩦ࡣලయ౛࡟ᑐࡋ࡚ṇࡢ㌿⛣ࢆࡶࡓࡽࡍࡇ࡜ࡶ㸪ᢳ㇟౛ࡀ᝟ሗ㔞ࡢᑡ࡞࠸༢ 

ㄪ࡞グྕ࡛࠶ࡿ࠿ࡽࡇࡑ㸪ᙧᘧⓗ࡞ᵓ㐀ࢆㄆ㆑ࡋࡸ

࡟⪃ᐹࡍࡿຊࠖࡀᥖࡆࡽࢀ࡚࠸ࡿࠋࡇࢀࡣᢳ㇟໬࡜ࡍࡃ࡞ࡿ࠿ࡽ࡜⪃࠼ࡽࢀࡿࠋࡑࡢ㐺⏝ࢆᣑ኱ࡍࡿࢫ࡛࢟ࣝ࠶ࡾ㸪Ꮫ⩦ࡢ㌿⛣ࡢ⌧ࡲࡓ㸪ᮏ◊✲࡛ࡣඛ⾜◊✲࡛ࡣྲྀࡾᢅࢃࢀ࡞࠿ࡗࢀ࡜ࡋ࡚㸪ᩘᏛᩍ⫱࡟࠾࠸࡚⋓ᚓࡀᮇᚅࡉࢀࡿ⬟ຊࡓ཯ᛂ᫬㛫ࢆ᪂ࡓ࡞ᣦᶆ࡜ࡋ࡚⏝࠸ࡓࠋࡇࢀࡣ㸪㌿࡛࠶ࡿࠋࡑࡢࡓࡵ㸪᜛ពᛶ࡜㌿⛣࡟㛵ࡍࡿ࢚ࣅࢹࣥ⛣ࡀ⏕ࡌࡸࡍ࠸ሙྜ࡟ࡣ㸪Ꮫ⩦ࡋࡓࡇ࡜࠿ࡽ㛵ಀᛶࢫࢆ✚ࡳ㔜ࡡࡿ࡜ྠ᫬࡟㸪ᑐ㇟⪅ࢆᩍ⫱⌧ሙ࡟⛣ࡋ㸪ࢆぢฟࡍࡲ࡛ࡢ᫬㛫ࡶ㏿ࡃ࡞ࡿ࡜ண ࡋࡓࡓࡵ࡛࠶Ꮫ⩦ࡢ㌿⛣࡟᭷ຠ࡞ᡭ❧࡚ࢆ⪃࠼ࡿᚲせࡀ࠶ࡿࠋࡿࠋࡋ࠿ࡋ࡞ࡀࡽ㸪ᮏ◊✲ࡢᡭ⥆ࡁ࡛ ᐃࡉࢀࡓ཯ ᮏ◊✲ࡣ㸪ᢳ㇟౛ࡢ㌿⛣ࡢඃ఩ᛶࡢ☜ㄆ࡜᜛ពᛶᛂ᫬㛫ࡣ㸪㌿⛣ࡢຠᯝࢆ♧ࡍ᭷ຠ࡞ᣦᶆ࡛࠶ࡿ࡜ࡣ࡜㌿⛣ࡢ㛵㐃࡟ࡘ࠸࡚ࡢ᪂ࡓ࡞どⅬࢆ♧ࡋࡓࠋ௒ᚋ㸪࠾ࡑࡽࡃゝ࠼࡞࠸ࡀ㸪㌿⛣ࡢࢫ࣒࣮ࢬࡉࢆ᳨ウࡍࡿୖグࡢㄢ㢟ࡀᐇ㦂ⓗ࡟᳨ウࡉࢀ㸪ᩍ⫱⌧ሙ࡛ࡢᩍᮦᣦᶆ࡜ࡋ࡚ᮇᚅ࡛ࡁࡿࡓࡵ㸪௒ᚋࡢ◊✲࡛ࡶᣦᶆ࡜㛤Ⓨ࡛ά⏝ࡉࢀ࡚࠸ࡃࡇ࡜ࡀᮇᚅࡉࢀࡿࠋࡋ࡚๰ពᕤኵࡋ࡞ࡀࡽྲྀࡾධࢀࡽࢀࡿ࡭ࡁ࡛࠶ࢁ࠺ࠋ ᮏ◊✲࡛ࡣ㸪᜛ពᛶ࡟ࡼࡿ㌿⛣ࡢ⏕ࡌࡸࡍࡉࡣぢᘬ⏝ᩥ⊩ࡽࢀ࡞࠿ࡗࡓࠋࡋ࠿ࡋ㸪㌿⛣ࢸࢫࢺࡀᢳ㇟ⓗ࡛࠶ࡿBrown, M. C., McNeil, N. M., & Glenderg, A. M. (2009).

ࡇ࡜࠿ࡽ⪃࠼ࢀࡤ㸪㌿⛣࡟㈉⊩ࡍࡿࡢࡣ NA ⩌ࡼࡾ

Using concreteness in education: Real problems,ࡶ A ⩌࡛࠶ࡾ㸪ࡑࡢ㏫࡛ࡣ࡞࠸࡜ண᝿ࡉࢀࡿࠋ⤫ィ

potential solutions. Child Development Perspectives, 3ⓗ࡟ࡣࡇࡢ᝿ᐃࢆᨭᣢࡍࡿ⤖ᯝࡣᚓࡽࢀ࡞࠿ࡗࡓࡀ㸪(3), 160-164.

࢖ࣛࢫࢺࡢලయᛶࡼࡾࡶࡑࡢ㌿⛣ຠᯝ࡬ࡢᐤ୚ࡀᙅDe Bock, D., Deprez, J., Van Dooren, W., Roelens, M., &࠸࡜ࡋ࡚ࡶ㸪᜛ពᛶ࡟ࡼࡿ㌿⛣ࡢྍ⬟ᛶࡣ௒ᚋࡶ᳨Verschaffel, L. (2011). Abstract or concrete examples in

ウࡍࡿᚲせࡀ࠶ࡿࡔࢁ࠺ࠋ

learning mathematics? A replication and elaboration ofᩘᏛᩍ⫱࡟࠾ࡅࡿ౛ࡢᛶ㉁ࡢຠᯝࡣ௒ᚋࡶᘬࡁ⥆Kaminski, Sloutsky, and Heckler’s study. Journal forࡁ◊✲ࡉࢀࡿ࡭ࡁෆᐜ࡛࠶ࡾ㸪ᐇ㝿ࡢᩘᏛᩍ⫱࡟ᛂResearch in Mathematics Education, 42 (2), 109-126.

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ᢕᥱࠖࡍࡿࢫ࢟ࣝࡶ㔜せ࡟࡞ࡿ࠿ࡽ࡛࠶ࡿࠋࡇࡢⅬRelevant concreteness and its effects on learning and࡛㸪ᮏ◊✲ࡣලయ౛ࡢ౑⏝ࢆྰᐃࡍࡿࡶࡢ࡛ࡣ࡞࠸ࠋ

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ᗘࡶ㔜せ࡛࠶ࢁ࠺ࠋణࡋ㸪࢖ࣛࢫࢺࡀ㌿⛣ࢆ㜼ᐖࡍKaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2008).

ࡿྍ⬟ᛶࢆ㋃ࡲ࠼ࢀࡤ㸪ᢳ㇟ⓗ࡞ᩘᏛⓗᵓ㐀ࡢᏛ⩦The advantage of abstract examples in learning math.

࡟㈉⊩ࡍࡿලయⓗ౛♧ࡸ౛♧᪉ἲ࡟㛵ࡍࡿ㐺ṇฎ㐝

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஺஫స⏝࡟ࡘ࠸᳨࡚ウࡍࡿࡇ࡜ࡀ㔜せ࡛࠶ࢁ࠺ࠋ㔠⏣ ⱱ⿱ (2012). ࣇ࢕ࣥࣛࣥࢻࡢ⟬ᩘᩍ⛉᭩ࡢຍ 2 Ⅼ┠ࡣ㸪ᐇ㦂ࡢሙࡸᐇ㦂ࡢᑐ㇟⪅ࢆᩍ⫱⌧ሙ࡟ῶἲࡢ࢖ࣛࢫࢺ࣮ࣞࢩࣙࣥ ᮾὒ኱ᏛᩥᏛ㒊⣖⛣ࡍࡇ࡜࡛࠶ࡿࠋᮏ✏ෑ㢌࡛ゐࢀࡓࡼ࠺࡟㸪᪥ᮏࡢせᩍ⫱Ꮫ⛉⦅, 38, 17-23.

୰Ꮫᰯ࣭㧗➼Ꮫᰯ࡛ࡢᩘᏛᩍ⫱ࡢ┠ᶆ࡟ࡣࠕ஦㇟ࡢ

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