text:
Viviana Carolina Llanos
email: vcllanos@exa.unicen.edu.arrotero@exa.unicen.edu.ar

María Rita Otero

Changes in the images and arguing from the mathematics textbooks for the secondary school in Argentina along 67 years

Keywords: Textbooks, mathematics education, secondary school

This work analyses the changes in the relationship between arguing and images from the mathematics textbooks for the secondary school in Argentina along 67 years. The textbooks have been published in the period 1940 thru 2007. The analysis is done by (N=137) textbooks based on three meta-categories in an inductive way. A factorial analysis of multiple correspondences was performed to find the main similarities and differences between the textbooks and to make a cluster analysis and one possible classification.

Introduction and conception of arguing

In 1994 an educational reform was performed in Argentina. The syllabus was changed and the scholar textbooks were adapted to the new educational system. The main modifications were realized in the properties of the images more than in the content. The characteristics of the text books, the ideas about argumentation and the characteristics of the images in the books are analysed. This research adopts an idea of arguing that emphasizes the relevance of the divergences between different points of view and the epistemological function of arguing, proposed by [START_REF] Leitão | Processos de construção do conhecimento: a Argumentação em foco[END_REF]. Different from other theories as [START_REF] Schwarz | Argumentation and mathematics[END_REF], [START_REF] Driver | Establishing the norms of scientific argumentation in classrooms[END_REF]; Leitão proposes that arguing has to be analysed based on three elements: "argument, argument against and response" in order to generate confrontation between argument and argument against, to achieve construction of knowledge and transformation of perspectives in the subject (response). These processes occur into face to face situations, or in negotiations of the different points of view with ourselves, in this case, when we are reading a textbook.

Methodology, categories of analysis and some results

A set of (N=137) mathematics textbooks is selected by means of purposive sampling techniques. The analysis was performed starting from a previous qualitative inductive categorization based on three meta-categories:

A-Characteristics of arguing. A1-Commencement of arguing: Questions or situation, which will be answered later; Definition, using to introduce knowledge; and Examples to formulate knowledge. A2-Type of arguing: Deductive formal, used deductive mathematics argument (definition, theorem, hypothesis, theory, demonstration, etc.); Deductive informal, they do not reach the formalism of the demonstrations; and Inductive that generalize knowledge from a single case. A3-Degree of arguing, cognitive conflict promoted by the text is analysed in three levels: High, books that generate explicitly confrontations, without solution in the text; Low, textbooks generate explicitly a cognitive conflict, solved later; and Absent, textbooks that inform without questioning.

Thematic Working Group 22

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B-Relationship between the images and arguing [START_REF] Otero | El Uso de Imágenes en Textos de Física[END_REF]: B1-Use of the image: Ornamental, images used with a decorative aim, not related to the content; and Argument, used as source of information, knowledge can be derivative. B2-Type of image: Mathematical representations, use mathematical systems of representation; and Non-mathematical representations, images not related with mathematical content. B3-Grammatical style of the images: Conceptual, represent relations and fixed characteristics between the represented elements; and Narrative, identify actions between objects that can represent a relation between them in the image. B4-Relationship with the "real world": Naturalist, images referring to the empirical world, detailed and complex; and Abstract, not referring to the world that we experience.

C-Characteristics of the textbooks: C1-Date of publishing: Period 1, 1940 thru 1973, Period 2 1974 thru 1994 and Period 3, after the reform, until the year 2007. C2-Educational level: refers to the educational level the textbooks. Level 1, students between 12 and 14 years old; Level 2, students between 15 to 17 years old; and Level 3, older than 18 years old. C3-Mathematical traditions [START_REF] Klimovsky | Las desventuras del conocimiento matemático: Filosofía de la matemática una introducción[END_REF]): Computational, emphasis in the resolution of problems and calculation with numbers; Axiomatic, present the mathematics demonstrations steps; and Structuralist: books that search of regularities that meet the same conditions.

Using this categorization, a qualitative description is made, which originated a first analysis. Then, the categorization is transformed in a group of nominal variables and modalities using Exploratory Data Analysis [START_REF] Lebart | Système SPAD[END_REF]. A Factorial Analysis of Multiple Correspondences allowed the selection of one possible classification in three classes. In addition, a test of randomness to analyse the reliability of the sample was performed using the statistical software SPAD.

The analysis explains changes in the images and arguing, given by: books that propose questions, or only definitions and examples, by the way to conceive and validate to mathematical knowledge, and mainly by the changes in the images and the relation between images and knowledge. The goal of most books seems to be informative. This explains the absence of questioning and discussing about several points of view, and the low level of arguing and conflict found within them.


domain: math shs

text:
Aleksandra Čižmešija

Ana Katalenić
email: akatalenic@foozos.hr

Željka Milin Šipuš

Asymptotes and asymptotic behaviour in graphing functions and curves: An analysis of the Croatian upper secondary education within the anthropological theory of the didactics

Keywords: Asymptote, asymptotic behaviour, anthropological theory of didactics (ATD)

published or not. The documents may come

Asymptotes and asymptotic behaviour in graphing functions and curves: An analysis of the Croatian upper secondary education within the anthropological theory of the didactics

Aleksandra Čižmešija, Ana Katalenić, Željka Milin

Introduction and methodology

Asymptotes and asymptotic behaviour play an important role in many traditional and modern mathematical disciplines and are supported by a rich and well-established abstract theory. Some basic aspects of these concepts can be utilized already in elementary mathematics as powerful tools in graphing and analyzing behaviour of elementary functions at infinity and near singularities and in graphing simple plane curves such as hyperbola. Therefore, this body of knowledge is a common part of upper secondary mathematics curricula worldwide. Here, we investigate its didactical transposition to the Croatian general upper secondary education. The survey is conducted within the theoretical framework of the Anthropological theory of the didactics (ATD), developed by Y. Chevallard [START_REF] Chevallard | The Didactics of Mathematics: Its Problematic and Related Research[END_REF][START_REF] Chevallard | Readjusting Didactics to a Changing Epistemology[END_REF]. The main idea of the ATD is to determine a relation RI(p,O) between a body of knowledge O and a person that occupies a position p in an institution I. Accordingly, mathematical knowledge and activities are described in terms of a praxeology [T, τ, θ, Θ], where its practical component, praxis, is represented with a task T and a technique τ and its discursive or theoretical component, logos, with a theory Θ and a technology θ. Research within the ATD should include relevant data about the relation RI(p,O) for all institutions I involved in the educational process i.e. the process of didactic transposition [START_REF] Bosch | Science or magic? The use of models and theories in didactics of mathematics[END_REF]. Questioning and understanding conditions and constraints on the relation RI(p,O) is necessary for setting up attainable, justified and significant educational interventions [START_REF] Barbé | Didactic Restrictions on the Teacher's Practice: The Case of Limits of Functions in Spanish High Schools[END_REF]. Hence, as relevant for our setting, we analyzed and compared the relations RB(p,O) and RS(p,O), where O is a graphical representation of an elementary function and a hyperbola in a square coordinate system, together with corresponding techniques regarding asymptotes, while institutions considered are two actual Croatian mathematical gymnasium textbooks B and the cohort of 40 the final, fifth year mathematics education students S at the largest mathematical department in Croatia. This is a part of a more comprehensive study regarding asymptotes and asymptotic behaviour in the Croatian pre-university education which also included the institution of academic mathematicians. The methodology included a praxeological analysis of the textbooks as representatives of the knowledge to be taught and three questionnaires with open-ended questions for the prospective mathematics teachers to provide an insight in related knowledge available to students as a potential taught knowledge. Based on this, a reference epistemological model (REM) is proposed and then verified and improved with scholarly knowledge gained from semi-structured interviews with two academics. Here, we focus on textbook topics, tasks, techniques and discourses related to O and on three particular tasks from the questionnaires administered: graphing and describing a simple rational function and a shifted exponential function from a real world problem, given by formulas, and graphing a hyperbola and its asymptotes.

Results and conclusions

The results of the textbook analysis show that graphing a function f pointwise, that is, by plotting some corresponding points (x, f(x)) and connecting them by a smooth curve dominates all other available techniques for graphing functions given by formulae. A technique of graphing a function or a curve regarding its properties recognized from a corresponding algebraic expression (formula or equation) occurs only in relation to the tangent and cotangent function and to a hyperbola, while the tasks on graphing polynomials and rational functions appear only as common practical activities of utilizing derivatives (calculus). Finally, a technique of transforming a prototype graph of an elementary function to get a graph of its composition with a linear function (by translations and dilations) is rarely implemented. Although textbook praxeologies of graphing functions often elaborate function properties and flow, discursive accents are set on establishing a function's domain, monotonicity and symmetry of its graph, neglecting its asymptotic behaviour. Asymptotes are seen relevant only for tangent and cotangent functions. Results of the questionnaire analysis completely reflect those of the textbooks. The students applied techniques and provided discourse to the same extent as it is given in the praxeological organization of the textbooks. Namely, their dominant techniques are drawing curves through their points and drawing graphs regarding function properties determined by using calculus, their chosen techniques are not the most efficient for the task in question, and asymptotic behaviour is available to them but not fully utilized in praxeologies relevant to graphing functions or curves. Considering this, it is our suggestion for the teaching practice and for an ongoing curricular reform in Croatia that: all relevant function properties should be emphasized when describing its behaviour; common properties of a function should be more related to its algebraic representation and utilized for its graphing; functions should be graphed by graph transformations, whenever fitted; and asymptotic behaviour should be more emphasized, adequately graphically represented and described by formal and informal mathematical discourse.

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