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øøøø(((Dl(((l(((ÿÿÿÿCOMBINATORIAL REASONING: AN ANALYSIS OF ELEMENTARY STUDENTS’ ERRORS
Tânia Campos (UNIBAN) HYPERLINK "mailto:taniammcampos@hotmail.com" taniammcampos@hotmail.com
Verônica Gitirana (UFPE) HYPERLINK "mailto:veronica.gitirana@gmail.com" \t "_blank" veronica.gitirana@gmail.com
Sandra Magina (PUC/SP) HYPERLINK "mailto:sandra@pucsp.br" \t "_blank" sandra@pucsp.br
Alina Galvão Spinillo (UFPE) HYPERLINK "mailto:alinaspinillo@hotmail.com" alinaspinillo@hotmail.com
Combinatorial reasoning has been indicated to be developed since the first years of elementary schools (Brasil, 1997). The study of combinatory has been introduced earlier mapping possibilities. According to Batanero, Navarro-Pelayo and Godino (1997), a person can solve a combinatorial problem by: (a) Intuition – randomly guesses; (b) mapping possibilities and (c) applying the formulas. Fischbein & Grossman (1997) shows that intuition on combinatorial problems has two forms: randomly intuition, that are not based on problems data and their structure and the intuition based on the sequential structures. Nonetheless, before developing combinatorial abilities, the child needs to recognize the principle of combination, and accepting it. Correia & Fernandes (2007) discusses the strategies of 9th school-year students on combinatorial problems. In the research they show that despite some student’s intuitive strategies being limited, the results show that these ideas have to potentially be a starting point to the study of combinatory. This paper reports a study aiming to map 8-11 years-old child’s error on dealing with combinatorial problems.
The combinatorial problems can be characterized to be the formation of n-vectors within M-elements, and are in combinatorial science, classified on: permutations, variations and combinations, depending on the possibility of repeating the elements to be combined. In the first years of elementary schools, the majority of problems exploited deals with formation of pairs choosing elements from two different sets. These problems are also classified by Vergnaud (1988) when studying the multiplicative structures as being a problem of formation of Cartesian Products, or a bilinear function. Among the Cartesian Products problems, Vergnaud classified them as Combination problems. He also differentiates the problems according to what is given and what is requested from students. The problem, for example: “Marta has 4 blouses and 3 skirts of different colors. She wishes to combine the blouses and the skirts to make cloths. How many cloths can she make?” Give students the numbers of elements and request them how many combinations can be done. Another kind of problem is composed by those that give the total of combination and the number of one of the elements to be used in the combination and request to students to calculate the number of the other problem. For example: “Exchanging blouses and skirts of different colors, Anna can make 20 different cloths. She has 5 skirts. How many blouses does she have?” Thus, classified in the multiplicative structure it seems to be essential to stratify the intuition strategies in child development toward perceiving the situation as correlated to multiplication.
This paper analyses errors committed by a hundred students from 8 to 11 years-old from 2nd, 3rd, 4th and 5th years of Brazilian elementary schools, particularly from the cities Recife and São Paulo. The two combination problems used above as example, classified by Vergnaud (1988), as Cartesian product - Combination were proposed to each child who solved them individually.
Children solved the problems in three different situations: (A) with paper and pencil, without researchers intervention individually in class; (B) with paper and pencil, with researchers intervention individually in a clinical study; (C) with manipulative material, with researchers intervention individually in a clinical study. Each child solved the problem in only one of these situations. Despite these papers are considering data from these different situations, it does not intend to compare them. It intends to bring light on child’s development on combinatorial reasoning.
The results showed a big percentage of errors, what led us to take a deeper view of these errors, on trying to propose a classification of these errors, as are discussed below.
ERROR 1: The child does not make any operation neither any kind of strategy to solve s/he answers by repeating a part of the problem or drawing blouses and skirts that can be the numbers given by the problem.
ERROR 2: The child uses an inappropriate operation with numbers that can or not be those given by the problem. It is more frequent for a child to use addition in the problem that requests multiplication and to use subtraction in the case of requesting division. The operation can or not be followed by drawings that represents blouses and skirts and lines connecting blouses to skirts. Nonetheless, there is no organization of these lines to allow them to build a combinatorial schema.
ERROR 3: The child solves the problems considering fixed combinations, s/he does not accept that a skirt/blouse can be used with more than one blouse/skirt. Once a cloth is formed, this cloth can not be used. In some cases, they use arrows to link blouses to skirts, when using paper and pencil. The arrows represent the one to one correspondence (a blouse to a skirt or a skirt to a blouse). We coined this error as fixed combination.
ERROR 4: The child solves the problems with more flexible combinations (not completely fixed). S/he accepts and one blouse that can be combined with more than one skirt. This suggests at first insight, even that an elementary one, of a one-to-many correspondence. Nonetheless, it is not an exhaustive, in the sense that each blouse used with all the skirts. The child accepts flexibility on the combination in order exhaust the other elements, in the sense that none blouse or skirt stay without being used. In fact, child thinking is still marked by the one-to-one correspondence. What differentiate this strategy to the one fixed combination é the fact to accept the Idea that a blouse can be combined to more them a skirt, demonstration a flexibility to combine. We coined this error as flexible combination.
All the error pointed above shows children who are still in the intuition strategies phases. Nonetheless, Error 3 and o Error 4 are qualitatively different from the other two kind of error identified, because they indicate reasoning that involve combination, what is clearly absent of the other kind of errors identified in this research. Error 1 and Error 2 does not involve any kind of combinatorial reasoning. Thus, this indicates a progress from the first two to the other two errors.
Another qualitative different can be observed between Error 3 and Error 4. While Error 3, the combination principle is based exclusively in a one-to-one correspondence, in Error 4, there are some indications of a first element of a one-to-many correspondence. Nonetheless, it is also important to observe that child who presented these errors were not consistent in all the problems. Some children are flexible for one problem, but use fixed combination to the other.
It is also important to observe that errors of building schemas have not been observed among these subjects, which does not allow exhausting to combinations, permitting that both skirts and blouses to be used more than once at the same problem. Error of mapping possibilities has not been observed.
REFERENCES
Batanero, C.; Navarro-Pelayo, V.N. & Godino, J.D. (1997) Effect of implicit combinatorial model on combinatorial reasoning in secondary school pupils, Educational Studies in Mathematics, v. 32, no. 2, pp. 181-199.
Brasil. Secretaria de Educação Fundamental (1997) Parâmetros curriculares nacionais : matemática. Secretaria de Educação Fundamental. – Brasília, MEC/SEF.
Correira, P.F. & Fernandes, J.A. (2007) Estratégias intuitivas de alunos do 9.º ano de escolaridade na resolução de problemas de combinatória. In BARCA, A. [et al.], ed. lit. – Congreso Internacional Galego-Portugués de Psicopedagoxía : libro de actas. A Coruña : Universidade, 2007. p. 1256-1267.
Fischbein, E. & Groosman, A. (1997) Schemata and Intuitions in Combinatorial Reasoning, Educational Studies in Mathematics, v. 34, n.1, pp.27-47.
Vergnaud, G. Multiplicative Structure. In Hiebert, H. & Behr, M. (1988) Reseach Agenda in Mathematics Education. Number Concepts and Operations in Middle Grades. Laurence Erlbaum Ed., Hillsdale, pp. 141-161.
We thanks to Adriana Batista who kindly give to some extracts of interviews from her master thesis, supervised by Prof. Alina Spinillo, in the Pos-graduation Program in Psychology of UFPE. We also thanks to CAPES (PROCAD No 0145050) that gave financial support to allow interchange between the authors on doing this research.
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