ࡱ> q`;bjbjqPqP4x::2 D D D X ` ` ` 8 \ X )1,!!!!!s"s"s"0000000$U2h4|0D )o"s" ) )0!!0--- )!D !0- )0--F D -!! 0:Y` +6-.00)1-95L,695-95D -s"+$:-e%a&s"s"s"00-^s"s"s")1 ) ) ) )X X X <$X X X <X X X  WHAT RELATIONS CAN WE MAKE BETWEEN PRIMARY TEACHERS CONCEPTIONS AND STUDENTS PERFORMANCE REGARDING TO ADDITIVE STRUCTURES? AN ANALYSIS OF TWO STUDIES IN BRAZILIAN SCHOOLS Eurivalda Santana Universidade Estadual de Santa Cruz-Brasil Sandra Magina Pontifcia Universidade Catlica de So Paulo-Brasil Irene Cazorla Universidade Estadual de Santa Cruz-Brasil Tnia Campos Universidade Bandeirante de So Paulo-Brasil ABSTRACT This paper presents a comparative picture between primary school teachers conceptions and students performance on the additive structures conceptual field. We have interviewed 103 Brazilian Primary School teachers, who were asked to elaborate four problems involving addition operation. Meanwhile, 1021 7-10 year-old students from 26 schools in Brazilian Public Primary School were asked to solve twelve additive problems. Results point out that teachers present the simplest problems related to additive structure categories, which are on composition and transformation prototypes, without diversifying problems into each type. Analogously, students presented best results in problems pertaining to these same prototypes; in more complex problems, students did not perform better even those with a higher scholarship degree. Results indicate the need to investigate what is happening and to intervene in continuous and initial courses for Primary School teachers, in order to allow them to develop knowledge about more complex additive structures in such a manner that they can, afterwards, contribute to students comprehensive vision of this conceptual field. Key-words: conceptions, competencies, additive structures. INTRODUCTION Compulsory School in Brazil dont have national standards, but is ruled by National Curricular Parameters (Parmetros Curriculares Nacionais) PCNs, that, since 1996, guide its teaching. To the first grades of the Fundamental Teaching (Primary School, 7-10 year-old students), mathematical content is organized in four groups: Numbers and Operations, Space and Shape, Magnitudes and Measures and Information Treatment. Numbers and Operations group starts with number concept (natural and rational) and the operations of addition, subtraction, multiplication and division. Regarding addition and subtraction (Additive structures), PCNs explicitly guide their teaching by using problem-situations. At this teaching level, teachers are generalists, meaning that they are responsible for the teaching of all disciplines, and, in general, they have undergraduate course in Pedagogy. Given the importance of the additive field in the architecture of mathematical knowledge and the teachers role in developing students mathematical competencies, we have investigated the possible relations between primary school teachers conceptions and students competencies in additive structures. CONCEPTUAL FIELDS THEORY Vergnauds Conceptual Fields Theory (1982, 1991, 1996) give us elements to analyse students learning difficulties and is a powerful tool for constructing situations (tasks) to be applied to teaching. This happens because this theory presents a pattern which is coherent if we intend to study complex competencies development and learning. To Vergnaud (1996), knowledge should be seen inside Conceptual Fields, which are defined as a set of problems or situations and a set of concepts, theorems, procedures and symbolic representations that make possible to analyse and to treat several kinds of situations. Due to the broaden diversity of concepts involved in conceptual fields, they encompass the knowledge that students acquire at mean and at long terms. In this way, Additive Structures Conceptual Field puts together situations and concepts related to addition and subtraction operations, which can be analysed as simple problems of relation between the whole and its parts (composition); transformations involving an initial and a final state (transformation) and comparisons involving a referent, a referee and a relationship (comparison). It is also possible to find problems classified as mixed by Magina et al. (2001) and that Vergnaud (1982) classifies in three categories: given two information, one searches for a third by means of a composition (composition of two transformations); given two static relations, one searches for the third, which is generated from the transformation (transformation tying two static relations) and, at last, composition of two static relations. The classification of situations in these six categories is directly derived from the kind of mathematical concept involved in each situation. In addition, from the element one searches in each of the three simple categories, Magina et al. (2001) classified situations in extensions. Table 1 shows examples of each category with its extensions; such examples are similar to the ones used in the instrument administered to students during the research. Table 1. Examples of three Additive Structures simple categories extension, similar to the ones we presented to students. ExtensionCompositionTransformationComparisonPrototypeCP Whole is Unknown: Ana has 4 white pens and 5 black pens. How many pens does she have as a whole?TP Final State is Unknown: Bete had 4 dolls. Daddy gave her 3 more. How many dolls does she have now?1st extensionC1 One part is unknown: Pedro spent R$12,00 to buy a ball and a notebook. The notebook cost R$ 8,00 How much did he pay for the ball?T1 Unknown Transformation: Joo had 6 balls. He won a few and ended up with 10. How many balls did he win?2nd extensionCA2 Unknown Referred: Cludio has 9 stickers and Vincius has 5 more than him. How many stickers does Vincius have?3rd extensionCA3 Unknown relation: Maria has 5 dolls and Telma has 8. Which one has fewer dolls? How many dolls she has less?4th extensionT4 Initial State is Unknown: Carla bought 2 books and ended up with 10. How many books did she have before?CA4 Unknown Referent: At the end of a marbles game, Artur ended up with 14 marbles. Knowing that Artur has 6 marbles more than Everton, how many marbles Everton ended up with?Analysis of students performance, when solving problems involving such situations, allow the understanding of competencies and abilities of which they make a scheme in their solutions. To Vergnaud (1987), students competencies are of major importance tools for complex learning description and analysis; which occur during a long period of time. He still states that [...] students competency can be fully traced by his or her actions in a situation. Hence, in face of a situation, students make their own solving actions schemes, meaning their competencies are mobilised while interpreting, choosing data and operations to be performed. CONCEPTIONS To Ponte (1992), conceptions are concepts mini-theories and are closely related to Professional practice, pointing out paths and basing decisions. In this way, it is possible to visualise a relationship between teachers conception and his practice in classroom. To Thompson (1982), relationship between teachers conceptions and decisions for actions are complexes ones; conceptions about Mathematics perform a meaningful, however subtle, role in determining what will be taught, by each teacher. Therefore, teachers manifested conceptions can meaningfully influence their work in classroom. METHODOLOGICAL PROCEDURES Aiming to analyse possible relationships between teachers conceptions and students competence, we have made two experiments. The first one had as target investigate, in ten schools from Brazilian Public Primary School, 103 teachers conceptions about additive conceptual field. They were asked to elaborate four problems about addition and subtraction operations, during a continuous course for in service teachers. Problems were analysed and classified as in Table 1. Also, we have counted the number of problems we had as well as the number of different ones, classifying them from zero to four. The second experiment was done with Brazilian Primary School 1021 students (7-10 years-old), in 26 public schools. The kind of instrument was paper and pencil and the students had to solve twelve addition and subtraction problems. Protocols were collected by a team of in service teachers, under researchers supervision. Instrument application was simultaneous, during 1,5 hour. Answers were categorised as right and non-right (wrong or in blank). DATA ANALYSIS 103 teachers have elaborated 378 (91.8%) problems in 412 possibilities (4 each teacher), which means that 8.3% was left in blank (Figure 1). Among the problems, the majority was prototype: 42.5% in composition and 32.5% in transformation. Figure 2 shows the number of elaborated problems and we could verify that 74.8% of the teachers has proposed four problems; nevertheless, just 54.4% of them has elaborated two different problems and only one has given four problems belonging to four distinct categories. Figure 1. Classification of teachers proposed problems.Figure 2. Number of proposed and of different problems.Figure 3 illustrates the number of elaborated problems by category and by extension. We observe that teachers have done two, three and even four problems of the same kind: composition and transformation prototype. The more extensive problems were just a few.  Figure 3. Number of proposed problems by teacher and classified by category and by extension. Figure 4 show evidences of a non-relationship between the number of proposed problems and time of practice in classroom, which means that teacher time in service doesnt contribute to propose diversified problems. We observe that the mean years of service stood at 15.2 and these varied between one and 38 years.   Figure 4. Number of proposed problems versus time in service. These results allow us to conclude that these teachers conceive as additive problems just prototypes ones and they use only composition and transformation type. This conception shows relationship with students performance, as we will see in what follows. We had 1021 students from 1st to 4th grade of scholarship. Analysing their performance, we could observe the best results in prototype problems and the worst ones in more complex extensions (Figure 5). Besides, what comes to attention in these data is the small difference in performance between students from these four degrees of scholarship, mainly in more complex extensions problems.  Figure 5. Rightness rates in problems, by category and by extension, and by student degree of scholarship. These results, together with teachers conceptions we have unveiled so far, seem to indicate that teachers are not working with additive structures more complex extensions, and this is the reason why students do not evolve in a more substantive way to master this field. FINAL CONSIDERATION Results we have found seem to point out teachers conceptions about additive field as a factor that explain why students do not encompass it. They also prove, in an indirect way, what we have observed in our experience with continuous courses for in service teachers: many of them do not even know this structures more complex extensions. At last, we bring to discussion that it is necessary to study, in a more precise way, correlations between a teachers work and his students performance, as well as to delineate qualitative researches in order to better understand the reasons why additive field is not extensively worked in classroom. REFERENCES Ponte, J. P. (1992). Concepes dos professores de matemtica e processos de formao. In: Brown, M.; Fernandes, D.; Matos, J. F. & Ponte, J. P. Educao Matemtica: temas de investigao. Lisboa, Portugal: p.189-239. Magina, S. et al. (2001). Repensando adio e subtrao: contribuies da Teoria dos Campos Conceituais. So Paulo, Brasil: PROEM. Thompson, A. G. (1982). Teachers conceptions of mathematics and mathematics teaching: three case studies. Unpublished doctoral dissertation, Universidade da Georgia. Thompson, A. G. (1992). Teachers beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.). Handbook of research in mathematics teaching and learning. New York, NY: Macmillan Vergnaud, G. (1982). A Classification of Cognitive Tasks and Operations of Thought Involved in Addition and Subtractibon Problems. In. Addition and Subtraction: a cognitive Perspective (pp. 39-59). New Jerssey: Lawrense Erlbaun,. Vergnaud, G. (1987). Problem solving and concept development in the learning of mathematics, E. A. R. L. Y. Second Meeting, Tbingen, Septembre. Vergnaud, G. (1996). A Teoria dos Campos Conceituais. In. Brun, J. Didctica das matemticas. Traduo por Maria Jos Figueiredo. Lisboa: Instituto Piaget. pp. 155-191.  Lecture of the Universidade Estadual de Santa Cruz and PhD student at PUC/SP having a scholarship granted by CAPES.  Lecture of the Universidade Estadual de Santa Cruz and Pos-PhD student at PUC/SP having a scholarship granted by FAPESB.  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