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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 )

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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.

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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.

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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).

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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.

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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).

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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).

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Research in Portugal Romani children mathematical predisposition, have been noticed by

teachers in elementary schools Benites (1997: 78) teachers involved in the investigation pointed out that mathematical reasoning as a characteristic of Romani children (Ferreira,2003; Pires, 2005; Candeia,2006)

Romani children were not fully integrated in school

Their subject matter preferred was Mathematics,

Their better grades was in mathematics They “posses a great capacity for

mental calculations”

Research in Greece Chronaki’s (2003) uses the concepts of learning identities and that of Roma funds of knowledge as resources for instruction to gain a better understanding of Romani children as mathematical learners

Stathopoulou and Kalabasis (2007) Romany children don’t attend consistently school and they are not fully integrated in school Low aptitude of Romany students in formal mathematics, although they come to school with important culturally acquired knowledge

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“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)

oral/ mental calculations

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Divisons in the “head” yes ... In the paper, not very much!

  Ferreira (2003)

M: -How many escudos are 4 euros plus 4,5 euros?

A: - 800...plus 900...it is...1700. It is not difficult.

..... M: - How much is 50 euros plus 62

euros? A: - 112 euros. M: - And 35 euros plus 30 euros? A: - 65 euros M: - How did you do this calculation? A: - 30 euros, plus 30 euros, plus

5 euros. ....

  Pires (2005)

Gi: -How much do you sell these glasses? Róg: - There are some at 15 euros, others 10? Gi: - If I asked you how much are 5 glasses ... Róg: - How much each? Gi: - 15 euros. Róg: - 15? I shell multiply 5 to 15 is’nt it ? Gi: - Yes… Róg: - 15 and 15 make 30, 30 e 30 …60 Gi: - 15 plus 15 … 30 and 30 Róg: - 60…70…75. Gi: - 75, very well, well done!

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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)

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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.

The following examples taken from Pires (2005) illustrates how Roger enrolled in third grade, calculates.

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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.

15 Jorge’ s notes

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:

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

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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?

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(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. …….

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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?

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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!

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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).

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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.

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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).

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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.

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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.

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Children’ relationship to mathematics education

• In general, Romani children like to go to school and recognize that school knowledge has having a role in their future • Some feel included in schools • They are aware that schools “do not care” about gypsy culture.

- their relationship with mathematics is good. - they have good grades. - they are good problem solvers. - Some children resist to change from oral to writing mathematics, namely to use typical algorithms.

• Although they recognize the importance of formal education it is not compatible with their way of live

• Children and their families consider that educational system/ teachers do not care about their culture and about their education (teachers consider that children and families do not care about school) • In general: - Their relationship with informal math is good - They are good solvers in the case they are permitted to use the cultural acquired cognition while hardly use to move to the use of formal mathematics

In Portugal In Greece

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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)

Some final Considerations

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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.

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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.

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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.

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