1. Arithmetical environments
semantically anchored
Father Woodland in Italy
becomes Zio Tobia (Uncle
Toby)
Carlo Marchini (Mathematics
Department of Parma University)

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. 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. 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. Premises
5

6. Premises
6

7. Premises
Pictures by D. Raunerova
word picture icon letter definition
mouse M
cat C =
goose G =
dog D =
7

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. 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. 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. Some questions I
Consider 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. 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. 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. 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. 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. The Italian version
•Starting from Hejný et al. (2006), during the school
year 2007/2008 a team of teachers of Viadana
Primary School, Rossella Guastalla, Maura Previdi,
Roberta Santelli, under my supervision, prepared a
sort of guide for presenting and exploiting the Italian
version of FW environment.
•In the school year 2008/2009 Guastalla followed
and improved that guide applying it in two grade 1
classes.
•In the school year 2010/2011 the Anna Frank
School of Parma experimented the same activity
with teachers Losi and Pompignoli.
16

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. 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. 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. 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 letter
topo (plural: topi) T
gatto (plural: gatti) G
oca (plural: oche) O
cane (plural: cani) C
20

21. The Italian version
•The Italian version makes easier
the contemporaneous presence of
the meanings of number as quantity
and number as magnitude
21

22. The Italian version
22

23. Examples from Italian guide
•The first three sheets are devoted
to a familiarization with the four
animals, their drawings and icons.
•Use of masks, dramatizing, posters
is required.
•The fourth sheet presents the
definitions of animals in the Italian
version.
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24. With animals pictures
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25. With icons
25

26. With numbers
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27. With Cuisenaire-Gattegno’s rods
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28. Preferred representations
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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. A theoretical difficulty
• The paper of Hejný et al. (2006) presents the
drawing
and the writings
{CCG} ~ {DM},
{CXX} = {DD}.
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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. 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#.
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33. Zio Tobia’s activity results
Assessment 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