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## Variation theory as a tool for teachers

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**Variation theory as a tool for teachers**Angelika Kullberg University of Gothenburg, Sweden School of Education University of Jönköping**Individual interviews with teachers about the planned lesson**Individually planned and enacted lesson (video recorded) Learning study 1 Background: Research projectaboutif and howteacherschangetheir teaching afterparticipating in learning studies Learning study 2 Learning study 3 LGK- project: 4 studies in total (2 in Mathematics and 2 in Science) Project leader: Prof. Ulla Runesson, University of Jönköping Individual interviews with teachers about the planned lesson Individually planned and enacted lesson (video recorded) Stimulated recall interviews with individual teachers Angelika.Kullberg@gu.se**Variation theory (somebasicideas) Marton & Booth, 1997;**Marton, Runesson & Tsui, 2004 • Learning is seen as the discernmentofaspectsthatyouhave not previouslydiscerned (cf. Gibson & Gibson, 1955). • “To discern an aspect, the learner must experience potential alternatives, that is, variation in a dimension corresponding to that aspect, against the background of invariance in other aspects of the same object of learning” (Marton & Pang, 2006, p. 193). • Therearenecessaryconditions for learning – criticalaspectsto be discerned. A criticalaspect is not the same as students’ difficulties, sincea criticalaspectsayswhat the students needtodiscern.**Lesson before the learning studies**1. Discussion about a task: “A person spent one third of his savings on a cd-player, and bought 3 cd for 165 cr. He paid 1975 cr. How much was his savings?” [The task is not solved] 2. Discussion in pairs about “what is an equation?” 3. Discussion about the meaning of the equal sign. E.g. 12·2=3·2·2·2was compared with 3·6≠18·2=36 4. The equation 3x+5=20 is solved with a visual representation. 5. The equations 3x+20=5 and 2x+3=3x+4 is discussed and found unsolvable for positive integers. 6. + 495=1975 is solved. (task 1) 7. A task about 3 persons ages, 3x+(x+5)+x=40 is discussed and solved. 8. Students work in pairs with making equations to given answers. Angelika.Kullberg@gu.se**“6·4” invariant,**answersvary (meaningofequalsign) 1.Discussion about the equal sign. Example: calculate the area of a triangle. 6·4≠ =12 6·4 = =24 6·4= 12·2 = 24 Lesson after the learning studies ”4·” invariant,answersvary 2. Multiplication with x, comparing 3+3+3+3 = 4·3 (is not 43) with 4x= x+x+x+x=4·x (is 4x). operation invariant,examples vary 3. Discussion about = = 3, = = 5, = 4, = x 4. Discussion about addition and subtraction. 3+4=72+3+4=7+2 2+3+4-5=7+2-5 Discussion about multiplication, comparing 2·3+4=7·2with 2·3+4·2=7·2 ”3+4=7” invariant,operations vary Angelika.Kullberg@gu.se**Introducingx in an equationwithoneunknown**”3+4=7” invariant, ”3” varies, becomesx Angelika.Kullberg@gu.se**Introducing 2x and x in a division**”3+4=7” invariant, ”4” varies, becomes a part ofx ”3+4=7” invariant, ”3” varies, becomes a part ofx Angelika.Kullberg@gu.se**Conclusions**Changes in teaching (video data) The teacheruses variation theoryas a toolto make aspectsofequationswithoneunknow explicit • In lesson 2 the teacher makes moredistinctions/contrasts. E.g. Between addition and multiplication (operations). • In lesson 2 the teacherbringsupcriticalaspectsofequations (interview data). • In lesson 2 the teacheruses ”variation and invariance” in a systematic way. E.g. 3+4=7 is kept invariant. • In lesson 2 the teacheroftenuseseveralexamples for the ”same idea” Angelika.Kullberg@gu.se**Lesson 2**The teacher: “…these [items] are more explicit in relation to the purpose. And I know that I can’t vary too much, because then I loose what I want to bring out. Because then students can get stuck at so many other things, they can see so many other things that I don’t want them to see at this moment, because that is not what they are supposed to learn this time” Angelika.Kullberg@gu.se