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# Zio tobia

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### Zio tobia

1. 1. Arithmetical environments semantically anchored Father Woodland in Italybecomes Zio Tobia (Uncle Toby) Carlo Marchini (Mathematics Department of Parma University)
2. 2. Premises• The starting idea comes from the reading of the paper:• Hejný M., Jirotková D., Kratochvilová J. (2006)Early conceptual thinking. Proceedings. 30th PME (Vol. 3, pp. 289-296). 2
3. 3. Premises• The paper abstract states: A pupil’s mathematical development is aimed at a procedural rather than a conceptual style of thinking. Both types are characterised and we illustrate the consequences which neglecting conceptual thinking can bring. We describe a fairy tale context, which enables us to investigate conceptual thinking, its diagnosis and development of pupils of Grade one. Action and clinical research was carried out and some mental phenomena describing the thinking processes of pupils in the given context were found. 3
4. 4. Premises• Father Woodland (FW) is a fairy tale figure who looks after different animals and organises tug- of-war games.• The weakest animal is a mouse (M). Two mice are as strong as one cat (C). A cat and a mouse are as strong as a goose (G). A goose and a mouse are as strong as a dog (D). Other animals are introduced in a similar way. Each animal is represented by a picture, an icon and a letter. 4
5. 5. Premises 5
6. 6. Premises 6
7. 7. Premises Pictures by D. Raunerovaword picture icon letter definitionmouse Mcat C =goose G =dog D = 7
8. 8. FW mathematical environment• From: Hejný, M., Jirotková, D., Kratochvílová, J. (2006). Early conceptual thinking, Proc. 30th PME (Vol. 3, pp. 289-296). Prague• This context is suitable for developing the following mathematical concepts and competencies: – early number sense – understanding of the difference between a quantity (expressed in units) and a number (expressed in pieces) – pre-concept of equations – pre-concept of divisibility, the lowest common multiple and greatest common divisor – conceptual thinking in pupils not only at the elementary level – solving methods of linear equations – solving of Diophantine equations• It is also a diagnostic tool enabling to characterise both cognitive and meta-cognitive styles of pupils. 8
9. 9. FW mathematical environment• An important feature of this environment is that it presents itself simply, and pupils accept without problems, the relational thinking connected with equality.• Literature documents that an understanding of equality as a relation is crucial to the devel-opment of algebraic thinking (Alexandrou-Leonidou & Philippou, 2007; Attorps & Tossavainen, 2007; Puig, Ainley, Arcavi & Bagni, 2007).• Here we focus on formal number sentences, building on the work of Molina, Castro & Mason (2007) and, in particular, on relational thinking – a term that Molina et al. (2007) borrow from Car-penter, Franke & Levi (2003 9
10. 10. FW mathematical environment• The student employs relational thinking if s/he• “makes use of relations between the elements in the sentence and relations which consti-tute the structure of arithmetic. Students who solved number sentences by using relational thinking (RT) employ their number sense and what Slavit (1999) called “operation sense” to consider arithmetic expressions from a structural perspective rather than simply a pro-cedural one. When using relational thinking, sentences are considered as wholes instead of as processes to carry out step by step.” (Molina et al., 2007, p. 925) 10
11. 11. Some questions IConsider the FW environment; in your opinion:• Q1) Which other mathematical topics could it inherently embed?• Q2) Which transversal cognitive competences could it help to develop?• Q3) Which school grades would it be suitable for? 11
12. 12. Some questions II• Q4) Is the ‘Father Woodland and friends’ tale present in your childhood folklore?• Q5) In case of negative answer, think about a substitute environment in your folklore which allows the same mathematical development. 12
13. 13. Some personal answers• From the structural point of view, this environment is a sort of formal algebraic system with equality: Alphabet → icons or letters. Term formation → juxtaposition, Atomic formulae → equality of two terms Axioms → definitions 13
14. 14. Some personal answers• Furthermore the presence of pictures and the tale itself give the opportunity to encapsulate also the semantics of this formal system.• The formal system axioms are not completely exhibited. The semantic suggests a hidden presence of addition (given by juxtaposition) which also allows (thanks to true formulae in the intended semantics) to obtain a richer algebraic structure 14
15. 15. Some personal answers• This setting, with explicit (sintax) and implicit (semantics) rules is a sort of ‘Eudoxian semigroup’ for ‘magnitudes’: – Addition: associative and commutative – Integer multiples of magnitudes, – Ordering relation – ‘Compatibility’ of ordering relation with addition.• These aspects would be suitable proposed as example of formal system for secondary school, but they are intuitively practised by children. 15
16. 16. The Italian version•Starting from Hejný et al. (2006), during the schoolyear 2007/2008 a team of teachers of ViadanaPrimary School, Rossella Guastalla, Maura Previdi,Roberta Santelli, under my supervision, prepared asort of guide for presenting and exploiting the Italianversion of FW environment.•In the school year 2008/2009 Guastalla followedand improved that guide applying it in two grade 1classes.•In the school year 2010/2011 the Anna FrankSchool of Parma experimented the same activitywith teachers Losi and Pompignoli. 16
17. 17. The Italian version• The fairy tale of FW is completely unknown to Italian pupils, therefore we had to find a suitable setting for our children.• Father Woodland becomes Zio Tobia, the character of a children song settled in a farm (a translation of the ‘Old Mac Donald Farm’ song)• FW’s tug-of-war becomes the animals feed: for example two mice eat as much as one cat. 17
18. 18. The old farm of Zio Tobia• The activity has been splitted in two temporal phases.• First school semester: establishement of the semantics – visit to a neighbouring farm – musical education with ‘The old farm’ song (rhythms, clapping hands) – Linguistic education animals names and calls – artistic education.• Explicit mathematical aspects were not involved in this first stage. 18
19. 19. The old farm of Zio Tobia• Second school semester: mathematical exploitation of the environment.• Use of the teachers’ guide for – Arithmetic learning – Ordering relation – Use of symbolism – Early concept of equations – Attention to the language development 19
20. 20. The Italian version• The examples of protocols I will show come from our school activity. A translation of the previous table could be useful to understand them better: word picture icon lettertopo (plural: topi) Tgatto (plural: gatti) Goca (plural: oche) Ocane (plural: cani) C 20
21. 21. The Italian version•The Italian version makes easierthe contemporaneous presence ofthe meanings of number as quantityand number as magnitude 21
22. 22. The Italian version 22
23. 23. Examples from Italian guide•The first three sheets are devotedto a familiarization with the fouranimals, their drawings and icons.•Use of masks, dramatizing, postersis required.•The fourth sheet presents thedefinitions of animals in the Italianversion. 23
24. 24. With animals pictures 24
25. 25. With icons 25
26. 26. With numbers 26
27. 27. With Cuisenaire-Gattegno’s rods 27
28. 28. Preferred representations 28
29. 29. Some questions IV• Q9) Focus on the two type of writings: – C = 2G – I thought a lot about how much they can eat. A mouse (1), a cat (2) a goose (3 because you must line them all) and a dog (4).” Are there differences in the use of numbers? 29
30. 30. A theoretical difficulty• The paper of Hejný et al. (2006) presents the drawing and the writings {CCG} ~ {DM}, {CXX} = {DD}. 30
31. 31. Some questions V• Q10) Are these types of drawing and writing commonly used in your schools for the representation of other mathematical concepts? Which ones?• Q11) How would you avoid the possible conflicts due to these representations? 31
32. 32. The corral• 1A Matilde and 1B Nicolò: “Instead of a corral we could just draw a circle”• 1A Filippo and 1B Davide: “Instead of the corral gate we could just draw some lines”• 1A Ahmed proposed for gates a symbol very similar to the ‘sharp’ button, #, on the mobile keyboard.• Tiredness suggested the use of letters or numbers instead of pictures or icons.• Finally, the accepted representation was like #C C G#. 32
33. 33. Zio Tobia’s activity resultsAssessment X is a (rational) number in the interval [0,10] which results from 10 individual tests. The sample of 39 pupils presents the following distribution Zio Tobia results 60% 51% 50% relative frequency 40% 30% 21% 20% 13% 15% 10% 0% X <4 4< X <6 6< X < 8 8<X Assessment 33