2. Ethnomathematics and knowledge legitimating
One of the major theoretical contributions of
Ethnomathematics is the focus on “learners”, on different
ways of to legitimate their knowledge and on the
possibilities to face outside of school learning in
relationship with learning inside school. (Domitte,2004 )
2
3. Ethnomathematics, informal knowledge and
mathemathematics education
i) how different cultural groups possess particular ways
of approaching mathematics
ii) the social, cultural and political nature of the variables
and processes involved in Mathematics Education
iii) the complexity of the articulation between the
mathematical knowledge based in primary culture
and that promoted by schools, highlighting the
dissociation of the school mathematics from daily life.
3
4. Ethnomathematics, mathematics education, and informal
knowledge
iv) The variability of mathematical ideas, practices and
concepts that exist in different cultural groups.
v)The mathematical strategies used to solve daily
problems that are posed within communities and
families the which are not known by schools
vi) That children who know and are involved in home-
based mathematical practices might not have
mathematical success in schools.
4
5. School’s role
The growing cultural diversity of school population in
Europe, poses new challenges to schools and to
schooling equity.
Schools, minority group, dominant groups
should avoid “cultural closure” (Moreira, 2007)
should involve in the recognition of different ways of
knowing
in order to share cultural element to proportionate
constructive interactions as a way to educate for
peace, respect for diversity and social justice (D’
Ambrosio).
5
6. The disconnection between the school mathematical
curriculum and students’ daily lives
school mathematics contextualization,
dialogical processes, “involve students in a permanent
problematization about their existential
situations” (Freire, 1985, p. 56),
to conduct the application of mathematics on the
contexts of students’ experiences and thinking.
6
7. In regard to Romani communities
Several international studies have shown:
Low formal education level in general.
Children presence in schools is irregular and absence for long period
of time is frequent.
earlier drop-out, more frequently for girls.
Schools have limited knowledge about Romani culture, namely:
schedules and holidays are not adequate to the Romani way
of life
There is not the integration of children’s Informal knowledge in
school activities and the Romani culture is not
represented in school materials
( Fraser, 1992/1995; Okely, 1983/1993; S. Roman, 1980).
7
8. In regard to Romani communities
Several international studies have shown:
Research in Greece and in Portugal have revealed
that schools and teachers seem to show little interest:
in what knowledge Romani students bring with them
and thus,
in how to build on this knowledge for classroom
teaching. (Candeia, 2006; Chronaki, 2005; Ferreira,
2003; Pires, 2005; Stathopoulou & Kalabasis, 2007).
8
9. Research in Portugal Research in Greece
Romani children mathematical Chronaki’s (2003)
predisposition, have been noticed by uses the concepts of learning
identities and that of Roma funds of
teachers in elementary schools
knowledge as resources for
Benites (1997: 78) instruction to gain a better
teachers involved in the investigation understanding of Romani children as
pointed out that mathematical reasoning as mathematical learners
a characteristic of Romani children (Ferreira,
2003; Pires, 2005; Candeia,2006)
Stathopoulou and Kalabasis (2007)
Romani children were not fully
Romany children don’t attend
integrated in school
consistently school and they are not
Their subject matter preferred was fully integrated in school
Mathematics,
Low aptitude of Romany students in
Their better grades was in mathematics formal mathematics, although they
They “posses a great capacity for come to school with important
mental calculations” culturally acquired knowledge
9
10. oral/ mental calculations
“oral mathematics is understood as
the mathematics practices that are
produced and transmitted orally,
not including written strategies.”
“What is at stake is the
understanding that numerical
aspects of a social practice are
inseparable from the cultural
setting itself.” (Knijink, 2002)
10
11. Divisons in the “head” yes ... In the paper, not very
much!
Ferreira (2003) Pires (2005)
Gi: -How much do you sell these
M: -How many escudos are 4 euros glasses?
plus 4,5 euros? Róg: - There are some at 15
A: - 800...plus 900...it is...1700. euros, others 10?
It is not difficult. Gi: - If I asked you how much are
..... 5 glasses ...
Róg: - How much each?
M: - How much is 50 euros plus 62
euros? Gi: - 15 euros.
A: - 112 euros. Róg: - 15? I shell multiply 5 to
15 is’nt it ?
M: - And 35 euros plus 30 euros?
Gi: - Yes…
A: - 65 euros Róg: - 15 and 15 make 30, 30 e
M: - How did you do this calculation? 30 …60
A: - 30 euros, plus 30 euros, plus Gi: - 15 plus 15 … 30 and 30
5 euros. Róg: - 60…70…75.
.... Gi: - 75, very well, well done!
11
12. Another example tells us about Gustavo, a ten years
old boy enrolled in third grade, who, after giving good
answers based on oral calculations to questions such
as: the double of 12, the double of 25, the triple of 63,
when the researcher asked him to solve 25+25 =
the child wrote 4010
Gustavo’s answer reflects his reasoning, which was
to add twenty plus twenty, and them five plus five.
Ferreira (2003)
12
13. The following examples taken from Pires (2005) illustrates how Roger
enrolled in third grade, calculates.
Róg: - I put 3 on each side
I put 2 on each side
I put 1 on each side
Gi: - And now?
Róg: - I red “bottom up”. 123 for each
.......and he circled around the result with his pencil.
13
14. Róger look at the operation 843: 2 = , and said:
Róg: - This one I do not know!
Gi: - Why?
Róg: - Because of the 3.
Gi: - Try to do it as you used to do the others counts. It is the
same …
Róg: - Well, I know it!
Róg: - 4 for each side.
Róg: - It rests 1 …It stays down as in the others counts!
Gi: - The counts that you did with your teacher?
Róg: - Yes
Gi: - It stays down ....in the remainder?
Róg. – Yes. 14
15. In the division, 369:3=, Jorge gave the answer right way: “123”. In regard to the
division 643:2=, Jorge took a few seconds more than in the previous situation
to give the answer: “321”, explaining that “it remainders 1”. When was asked,
by the researcher, to explain his reasoning and to try to write down his thinking,
he wrote the following notes:
Jorge’ s notes
15
16. Romany students in Greek school context
and the use of informal cognition in
problem solving
This incident has happened in a 1st grade
Romany class. The students’ age
although arranged from 7 to 12 years
old. The students are called to solve a
typical problem of division (the have not
taught the typical algorithm). Students
manage to invent their own algorithms
17. Basilis wanted to help his father to distribute apples
in crates, which his father had got from the
vegetable market. All the apples were 372 kg and
every crate hold 20kg. How many crates does he
need in order to put in all the apples?
17
18. (Apostolis was drawing lines on his desk: for every crate one line).
R: please, tell me Apostolis what are you doing here?
A: 10 crates Miss.
R: How many kilos do the ten crates hold?
A: 20 kg every crate.
R: So….
A: Well, 20, 40, …….180, 200.
R: And how many are there?
A: 372
Cr: I am thinking Miss….
J: (He continues) 220, 240, ….
R: You Cris, what are you doing;
Cr: On this hand 72, the 60
E: What sixty; you mean sixty crates;
Cr: I don’t mean crates, 3 crates.
……. 18
19. R: How did you find it; (at the same time Apostolis and John
continue to step by 20 up to 372).
Cr: I said 20 (he shows for every crate one finger) and 20 and
20, 60 and the rest are 12. I get for these (and shows the hand
he imagines that he has the 300 kilos) 8 more, so I have 4
crates.
……..
Cr: I get from the 300 the 8, 4 crates miss, 8 and 12 miss the
rest are…….. 302. No they are ….
A: 250.
Cr: Wait! 292.
R: Bravo Cris. You had better write down the number so that
you don’t forget it.
Cr: I get from the 292, the 20, 5 crates, and the rest 272. Is it
ok miss
………..
J: Should I also do the same miss? 19
20. Cr: 10 crates and the rest are 172. I get some more and they
become 152. I am correct (with self-confidence).
J: look at him miss, he is doing them, he is doing them!!! (with
admiration).
Cr: I get one more. I have 12 crates and the rest are 132. Now I
get these 20 and the rest are 112, and I have 13 crates, all right?
A: All right.
Cr: Then, miss the rest are 112. Am I right; from the 100…. What
I have done now, I am confused.
E: You are here at 112, you get the 12…
Cr: And I get 8 more from the 100, and now I have 92.
A: look at it! Look at it!
20
21. Cr: I put one more here (he means one little cube, he used
for corresponding the crates)
…………….
A: Put one more crate, the rest kilos are 12
Cr: Twelve
……….
Cr: 1,2, 19 (he counts the cubes)
E: So, we needed 19 crates. What did you find?
A: 19
E: Could you show as your one solution;
A: (Corresponding every line to 20 kilos) 20, 40, ……
B: 100, 120, ….360, 380.
E: This last crate is going to become full?
All together: no
(Chris used continuant subtraction. It must be noted here
that the students didn’t know the algorithm of division, as
they are students of first grade level. What they had been
taught were simple operations with number up to 20). 21
22. Relationship between mathematical knowledge and
the Romani way of life (cultural context)
The main base of Romani children’s learning processes continues to
be realized inside their involvement in families and communities
activities and is directly connected to their cultural context.
Their cultural context consist of the following elements:
Semi-nomadic way of life with directs consequence on their schooling.
The socio-economic organization which has a family based model, so
children are involved in their families’ business and through a
horizontal way of teaching they become familiar with doing oral
calculations.
22
23. Cultural context and mathematical cognition (positive
effect)
Learning is done by :
observation
reproduction
of what they hear and see participating in family’s business
(and the corresponding practices).
They learn in a horizontal way and reproduce the
knowledge of their social group.
language’s orality—one of main cultural elements—is
related with their ease in oral calculations
(Cadeia, 2006; Ferreira, 2003; Fraser, 1992/1995; Okely, 1983/1993; Moreira e Pires; 2006; Pires, 2005; S.
Roman, 1980; Stathopoulou, 2005).
23
24. Cultural context and mathematical cognition (negative effect)
Semi-nomadic way of life has as consequence the time of starting
school and the inconsistency in attendance.
They are a minority and marginalized group with limited expectation
of education depending on their cultural fund.
24
25. Romany students in school context
The research has revealed, in both countries, that
although these children’s through their every day
experience acquire knowledge and agility towards
calculus using their own algorithms this knowledge is not
considered in classes.
Through these episodes it is revealed that children
possess own strategies—using oral calculations-- to solve
mathematical problems.
Children, in case they feel free– use their cultural
knowledge to solve mathematical problems in school.
25
26. Children’ relationship to mathematics
education
In Portugal In Greece
• Ingeneral, Romani children like to go • Although they recognize the importance
to school and recognize that school of formal education it is not compatible
knowledge has having a role in their with their way of live
future
• Some feel included in schools • Children and their families consider that
• They are aware that schools “do not educational system/ teachers do not care
care” about gypsy culture. about their culture and about their
education (teachers consider that children
and families do not care about school)
- their relationship with mathematics is
good. • In general:
- they have good grades. - Their relationship with informal math is
- they are good problem solvers.
good
- They are good solvers in the case they are
- Some children resist to change from
permitted to use the cultural acquired
oral to writing mathematics, namely to
cognition while hardly use to move to the
use typical algorithms.
use of formal mathematics
26
27. Some final Considerations
As Gerdes (1996a) notes, teachers education should
include training
“to investigate ideas and practices from their own
cultural, ethic and linguistic communities and to search
ways of constructing their teaching from them on (…)
and to contribute to the mutual comprehension, the
respect and the valorization of (sub) cultures and
activities” (p. 126)
27
28. Schools- mathematics education
The weakness of the educational system to be designed
or at least to be adaptable for Romany students --and
generally for students with cultural diversity– is that
formal education ignores or has the contempt for the
cognition children acquire through their everyday
context.
28
29. Schools- mathematics education
so, what is needed:
• to develop ways of knowing the social places their institutions,
problems, projects and social practices.
• to work out processes that promote students self esteem
• to promote methodologies that account for different learning
styles and for each student particular knowledge.
29
30. Further Research
To investigate:
• home based literacy and numeracy
practices
• The presence of numeracy practices in
the Romani cultural texts
• other aspects of the Portuguese/ Greek
Romani ethnomathematics
In conclusion, to promote an education based
in the respect of human rights, it is necessary
to bring inside the school the culture of
students, so that they feel being accepted,
being respected and being valued.
30
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