Performance of first-year undergraduate students attending exact sciences courses in problems of the additive conceptual field

This paper, which is part of a research based on the theoretical framework of the Theory of Conceptual Fields, investigated the performance of first-year undergraduate students attending Exact Sciences courses at the State University of Maringá, Brazil, with regard to the solution of problems in the additive conceptual field. Results obtained by these students evidenced some types of reasoning involved in additive structure that stand as an obstacle to the learning of Mathematics and may interfere with their future studies. Among the three classes that constitute the additive conceptual field proposed by Gérard Vergnaud, transformation problems had the highest rate of unsatisfactory results. However, errors committed by the students were detected in all problems of the diagnostic test. These results underscore the need for a broader discussion in academic environments that takes into account the difficulties in the test with regard to reasoning involving additive structure so that changes in the curricular structures of Exact Sciences courses may be fostered.


Introduction
The Theory of Conceptual Fields (TCF) by Gérard Vergnaud is the theoretical framework founding this research whose objective is to investigate the performance of students admitted to Exact Sciences courses at the Universidade Estadual de Maringá (UEM) in problems of the additive conceptual field.
An analysis of operational invariants presented by these students in the resolution of such problems was carried out in Kato et al. (2013, p. 33, our translation), pointing some concern regarding their interference with "[…] failure and dropout rates in undergraduate courses" 1 .
In this sense, this work used the same diagnostic instrument proposed in Kato et al. (2013) and was conducted with 309 freshman students attending Exact Sciences courses at the UEM in 2012, concerning the additive conceptual field, according to three classes -composition, transformation and comparison -proposed by Vergnaud (2009b).
This study goes beyond the mere mensuration of the knowledge of students attending Exact Sciences courses in college, because it contemplates a broader discussion about possible initial obstacles that may interfere with the performance of these students during their undergraduate studies, having dropout as the main consequence.
The concern with the permanence of students in Exact Sciences courses at the UEM has already been the object of a study conducted by Garcia et al. (2000) with first-year licentiate students, pointing that Higher Education dropout means an escape, failure or even a search for new horizons, and interferes with both the progress of disciplines and with the pedagogical project of courses, which emphasizes the importance of this discussion within the academic context.
Dropout rates in Exact Sciences courses have been turning the focus of some studies to a student's required skills, such as reading, interpretation, logical reasoning, etc., which interfere with his or her learning process (SILVA FILHO et al., 2007;CURY, 2008).
In this context, and also considering other studies about the reasoning involved in the additive conceptual field, like those by Magina et al. (2008), Santana (2010) and Vergnaud (1982), this work presents an overview of the performance level of first-year students attending Exact Sciences courses at the UEM concerning composition, comparison and transformation problems, according to the categories proposed by Vergnaud (2009b).
The questions proposed in the diagnostic instrument presented in Kato et al. (2013) contained six (06) problems, contextualizing situations of the everyday life of students that evoke different types of reasoning of the additive structure, because it is necessary to consider different degrees of complexity in order to provide an overview of how students master different situations of the additive conceptual field.
The meaning of the situation for an individual seems to be one of the decisive factors for him or her to have a good performance in a given situation.Actually, this meaning is intimately linked to the variety of schemes that he or she has in his or her repertoire, which makes it possible to solve a given situation with greater or lesser competence (SANTANA, 2010, p. 139, our translation)2 .
The relevance of this study is not restricted solely to the academic community from the UEM, environment where the sample was selected, because we believe that the discussions deriving from the results obtained will contribute to inspiring future researches about skills and competences required for an individual to be able to play his or her role as a professional satisfactorily.

Theoretical foundations
According to Vergnaud (1993), the TCF is a psychological theory on the process of conceptualization of reality and seeks to describe and analyze the mid and long-term progressive complexity of mathematical competences that students develop inside and outside the school (VERGNAUD, 2009a).
When the author highlights the progressive complexity of mathematical competences, he refers to the development and learning that students go through when facing problematic situations.Such complexity depends on the enunciations of situations, on the students' cognitive structure, on the context involved, on data numerical characteristic and on how it is presented to students.
The comprehension of a concept by a student does not happen when he or she is confronted with one single situation; after all, a concept does not derive only from one situation.Each situation always involves more than one concept.One single concept or an isolated situation does not make up the entire process of acquisition of a certain piece of knowledge, because mathematical concepts draw their senses based on a variety of situations.Vergnaud (1993, p. 11, our translation) describes two main ideas on situations: That of variety: there is a great variety of situations in a given conceptual field; situation variables are a means to construct systematically the set of possible classes; that of story: the knowledge of students is elaborated by situations that they face and have mastered progressively, especially for the first situations likely to give a sense to concepts and procedures that one intends to teach them3 .
The additive conceptual field is constituted of different situations containing one or more addition or subtraction operations, besides concepts and theorems necessary for the mathematical reasoning to be applied in these situations (VERGNAUD, 1993).
According to Magina et al. (2008, p. 19, our translation), the additive conceptual field can be categorized according to types of reasoning required for the resolution of different situations.This categorization aims to offer a theoretical structure that may help teachers to understand the meaning of different symbolic representations of addition and subtraction, and to serve as a basis to experiences concerning this mathematical process in the classroom.The theoretical reason for distinctions in categorization is of both psychological and mathematical origin4 .
To master additive structures implies to be capable of solving several types of problematic situations, which means that to know how to operate a numerical calculation does not suffice.Magina et al. (2008) state that, behind a simple operation like 3+5, there may be problems so sophisticated that many students might have trouble solving it.Moreover, the ways to organize information of situations differ from each other, although the same arithmetic operation is used to solve them.Santana (2010) and Vygotsky (1994) stress the relation between the role of social interaction, of language and of symbolization as the mastering of a concept by students progresses.In face of this, it is worth highlighting the importance of teachers as the mediators of teaching and learning processes that favor situations that enable students to develop schemes in the zone of proximal development.
The conceptual field of the additive structure is composed of concepts of cardinality, of measure, of temporal transformation (gaining and losing), besides relations of quantified comparison, of binary composition of measures, of composition of transformations and relations.
In this work, the problems used in the diagnostic test belong to the composition, the transformation and the comparison classes.
The composition problem class comprehends situations taken wholly and partly.This means to say that the problematic situation comprises two or more parts to identify the value of the whole.This composition may occur between measures, transformations and relations.
The transformation problem class involves a temporal idea, in which at times one loses, at times one gains, that is, transformations may be negative or positive.In this way, they are dynamic relations that establish relations between initial and final amount.Transformations occur between measures and between relations.
The comparison problem class occurs between measures, when one relation (comparison) links two measures.
Figure 1 displays the forms used to represent the relational calculus necessary for the resolution of the additive field problems.
The legends of Figure 1 will be used in the representation of the relational calculus of each one of the problems in the three classes.

Methodological procedures
This study counted with the participation of 309 students attending the first year of courses offered by the Center of Exact Sciences at the UEM, namely Mathematics, Physics, Chemistry, Statistics, and with students from the Department of Sciences of the Regional Campus of Goioerê.To carry out such a procedure, we used a diagnostic test made up of six problems elaborated by Kato et al. (2013), approaching different situations belonging to the additive conceptual field, and that contemplated the three major classes -composition, transformation and comparison -proposed by Vergnaud (2009b).
The methodological conception used in our work has an empirical and analytical nature with a descriptive design, being similar to the proposal by Campbell and Stanley (1979) referring to an experiemntal design that uses a single group of subjects, which answered the diagnostic test on the same day, but at different times, due to different courses and shifts.Table 01 describes the amount of participants in this research attending each one of the courses involved.
Data was collected from questionnaires in which the participants' private information was not identified.In this way, the individuals that were objects of this research had their privacy preserved and, after being informed about the objectives of the study, accepted to participate by signing an informed consent form allowing the use of their questionnaires in our work.
Considering our objective to present an overview of the performance level of students attending the first year of Exact Sciences courses at the UEM concerning composition, comparison and transformation problems, we present in Figure 2 the problems used in our diagnostic test, extracted from the work by Kato et al. (2013), as well as the relational calculus necessary to their resolution, and respective schemes.
P1, P4 and P6 are composition problems.This type of problem is characterized by the fact that two measures are composed so that they make room for another measure (P1), or two transformations that compose each other, resulting in a third transformation (P4), or by the fact that two relative states are composed so that they make room for another relative state (P6).
P3 is a comparison problem and is characterized by presenting a relation that links two measures.
P2 and P5 are transformation problems.P2 gives the initial (I) and the final (F) states and asks for the transformation occurred in the passage from a state to the other.P5 provides in its enunciation a relation (R1), a transformation (T) and asks for the relation (R2), based on R1 and T.

Analysis of results
In order to discuss the performance presented by all 309 participants of the research, we built four tables that describe the satisfactory and unsatisfactory results obtained in each class of problems.
Table 2 displays the results obtained in composition problems P1, P4 and P6.

Form Name Represents
Rectangle A measure, that is, a natural number.
Ellipse A transformation or a relation, that is, a relative number.

Horizontal curly brackets
Composition of elements of equal nature.

Horizontal arrow
A transformation or a relation between elements of equal nature.P4: Pedrinho's mother put some oranges in a basket for him to take it to his grandmother.On the way Pedrinho ate five oranges and felt worried, so he stopped by Mr. Raul's farm and picked more oranges.When he arrived at his grandmother's house, he counted the oranges in the basket and noticed that there were six more than the amount that his mother had put in the morning.How many oranges did Pedrinho pick at Mr. Raul's farm?In the category of composition problems, P4, of composition of transformations, was the problem which the students had more trouble interpreting in all courses investigated.In our view, the absence of the initial state in the problem, 'Pedrinho's mother put some oranges in a basket…' broke with the canonic form of resolution with which the students are familiar.Conne (1979apud VERGNAUD, 1982) observed that most of the procedures and explanations that students use during the resolution of problems can be interpreted based on the functional model: 'Where do I start?What do I do next?(addition or subtraction)?',but this model does not work for all types of situations, because it requires the identification of the starting point before beginning the resolution of the problem.
To that author, when the students force the resolution of a problem while supported on the functional model they invert the order of transformations or even make changes in gains or losses and vice versa, hence the cause of many errors in this type of problem.Such procedures were observed in the resolution of P4, as described in Kato et al. (2013).Table 3 shows the results obtained in said problem in comparison to P3.
Most of the students did not show difficulties when solving P3, and the incorrect results were a consequence, mainly, of a mistaken interpretation of 'how much one owes to whom'; in this case, the problem says: ' Although the phrasing of the problem indicates debts (losses), the resolution is given by addition, which, according to Vergnaud (1996), is a complexity factor involved in additive reasoning.Table 4 displays the results obtained in transformation problems P2 and P5.
Transformation, measure problems, P2, and relation problems, P5, were those that presented the highest rate of unsatisfactory results in the test, possibly because they involve temporal transformations, which are dynamic and, according to Vergnaud (1982), not adequately represented by natural numbers.Marthe (1979apud VERGNAUD, 1982) states that problems that have opposite signs are generally harder than problems that have all magnitudes with equal signs.This author's work comes to the conclusion that the comprehension of the conceptual field of additive structures requires a long period of time and, in general, is not completely developed until 15 years of age.
The results presented in Tables 2 and 4 show that among all courses investigated at the UEM -Sciences, Statistics, Physics, Mathematics and Chemistry -, errors have been detected in at least one of the problems of the diagnostic test.Such errors are directly related to the mastering of the types of reasoning involved in each class of problems.
The mastering of a conceptual field does not occur in a few months or years.On the contrary, new problems and new properties should be studied throughout several years if we want students to master them progressively.Trying to avoid conceptual difficulties serves for nothing; they are overcome as they are found and faced, but this does not happen all at once (VERGNAUD, 1982, p. 401, our translation) 6 .In Table 5 we present unsatisfactory results obtained by course, in composition, comparison and transformation classes.We highlight that the total of answers (T) is given by the number of students in each course multiplied by the number of problems in each class.The number of errors (E), in turn, is the amount of wrong answers inside this total.
Table 5 evidences that transformation problems had the highest rate of errors inside the investigated group, and that comparison problems had the lowest rate.
According to Vergnaud (2009b), the notion of logical complexity involves the hierarchy of different logical objects of which the problems treat (population, debt, etc.), of the different properties of these objects in the context of the problem and of the different classes of problems in which these properties are used (composition, comparison and transformation problems).
To Vergnaud (2009b, p. 60), "[…] innumerous relations of the real world are, in fact, dynamic relations, in the sense that they link successive states of reality rather than simultaneous elements of reality" 7 .Such dynamic relations are involved in transformation problems.
In the case of temporal transformations, for instance, at times one loses, at times one gains.These relations can be dynamic, for example, Ana has 3 caramels and gained 5, so now she has 8 caramels, or static, for example, Ana has 5 caramels and Tina has 3 caramels, so together they have 8 caramels.

Final considerations
Situations involving types of reasoning of the additive structure are experienced by individuals from their very first years of schooling; however, there is a diversity of contexts inside a single situation that interfere with the degree of complexity of the relational calculus involved.According to Vergnaud (2009b, p. 212, our translation), "[…] the greater or lesser facility of the necessary numerical calculation and the order and presentation of information in the problem" 8 also contribute to the performance of students during the resolution.
In this sense, we understand that a student learns through actions realized when he or she interacts with problematic situations, and that competence is directly related to successive degrees of complexity; the success in mastering it may require a long period of time, whether in school activities or not.
Our research pointed that the development of types of reasoning involved in the additive structure of freshman students attending Exact Sciences courses is still under construction, which corroborates the affirmations by Vergnaud (2003) that competences, necessary for one to master the additive conceptual field should be developed throughout the school and academic life, in all areas of knowledge.
The focus of this article is on analyzing the amount of students that came to the right or wrong answers to the problem but, mainly, on investigating the performance of first-year students attending Exact Sciences courses at the State University of Maringá (UEM) regarding problems of the additive conceptual field that may interfere with the continuation of their studies in college.
Evidently, the results presented in Tables 2 and 4 do not implicate nor justify dropout or failure levels of students attending Exact Sciences courses, but are an indication of the importance of a more accurate study about their possible causes, as pointed out in the work by Garcia et al. (2000), which should promote discussions and changes in the pedagogical sphere of these courses.
The presentation of results by course does not have a comparison intention nor seeks to infer any assumptions about the first-year students, but aims to point possible difficulties students face when it comes to composition, transformation and comparison reasoning, which may become an obstacle to learning, due to the specificities of each field of knowledge.However, the discrimination of the courses investigated aspire to draw the attention of teachers who are directly involved about this issue, allowing them to carry out, by means of their teaching activity, interventions that favor a learning that is not limited only to the student, but that also makes them adopt an attitude as that of a researcher of their own practice (SANTANA, 2010).Based on this observation, we infer that changes, whether in the pedagogical projects of courses, whether in the progress of the disciplines, are necessary to favor the development of the cognitive structure and of knowledge, allowing students to expand their skills and competences to solve more complex problems that might arise in their academic and professional journey, in which there is some interference by the additive field.

P1:
According to the demographic census of the Instituto Brasileiro de Geografia e Estatística [Brazilian Institute of Geography and Statistics] (IBGE), in 2011, for the first time in Brazil white people accounted for less than half of the total population.Among the 190,755,799 inhabitants registered in that year, 99,697,545 identified as being black, mixed, yellow or indigenous.What is the number of whites?Relational calculus: Measure 1 (M1) = 190,755,799; Measure 2 (M2) = 99,697,545 Composition: M1 -M2 = 91,058,254 P2: João will travel from Maringá to Campo Grande next Saturday, on a flight departing at 11h10min, and will arrive at 11h20, local time.What is the expected duration of this trip, knowing that Campo Grande's time zone is 1h behind Maringá?Relational calculus: Initial State (Ie) = 11h10min; Transformation (T) = -1h; Final state (Fe) = 11h20min -(11h10min -1h00min) = 1h10 min P3: Carlos lent R$300.00 to Antônio.André has a debt with Carlos too but he owes R$250.00 more than Antonio.How much André owes?Relational calculus: Measure 1 (M1) = 300; Relation (R) = + 250; Measure 2 (M2) = M1 + R = 550

Table 1 .
Distribution of the subjects investigated.

Table 2 .
Results obtained in composition problems.
Source: This research's instrument for data collection.

Table 3 .
Results obtained in the comparison problem.

Table 4 .
Results obtained in the transformation problem.

Table 5 .
Amount of unsatisfactory results by course.