An institutional conception about the derivative in the course Mathematics I at Universidad Tecnológica de Pereira

RESUMEN: Este artículo reporta resultados de una investigación que indagó por los significados institucionales del objeto matemático derivada en la Universidad Tecnológica de Pereira en el periodo 2009-2014, referidos en el primer examen parcial unificado de derivadas. Se identificaron tanto configuraciones epistémicas como contextos preferenciados por el grupo de profesores quienes diseñan las pruebas unificadas.ABSTARCT: This article reports the results of a research on the institutional meanings of the mathematical object derivative carried out at Universidad Tecnológica de Pereira during the period 2009-2014. Epistemic configurations as well as contexts preferred by the group of teachers who design the unified tests were identified

Elementary algebraic reasoning : proposals for the classroom

RESUMEN: En este artículo se aborda el problema de la enseñanza de álgebra en la escuela primaria. Se discute lo que se entiende como “early álgebra” y se presentan varios enfoques sobre la introducción del álgebra en la escuela elemental. En correspondencia con los enfoques, se discuten tanto algu- nas tareas de razonamiento algebraico elemental que pueden ser implan- tadas en el aula, como algunas de las características algebraicas atribuidas. Finalmente, se presenta una propuesta para asignar niveles o grados de algebrización a algunas tareas.ABSTARCT: He problem of teaching algebra in elementary school is discussed. It discusses what is meant by “early algebra” and various approaches to the introduction of algebra in elementary school are presented. In correspon- dence with the approaches, both some elementary algebraic reasoning tasks that can be implemented in the classroom, as some of the character- istics attributed algebraic discussed. Finally a proposal to assign levels or degrees of algebraization some tasks is presented

Preservice elementary teacher´s thinking about algebraic reasoning

ABSTARCT: His paper reports on the results of a research about the comprehension of elementary algebraic reasoning exhibited by preservice elementary teachers. Such comprehension is studied by means of the elementary algebraic tasks that preservice teachers propose to be worked out by primary pupils, as well as their conceptions about including elementary algebraic reasoning in primary school curriculum. Based on the analysis of the tasks it stands out the procedural and numerical conception exhibited by the preservice teachers. Additionally they conceded little importance to the relational aspect of the equal sign. Some implications for teacher education are discussed

Comparing the Didactic-Mathematical Knowledge of Derivative of In-Service and Pre-service Teachers

Background: The knowledge that a mathematics teacher should master has taken an increasing interest in recent years. Very few studies focused on comparing didactic-mathematic knowledge of in-service and pre-service teachers aimed at identifying features of the teachers’ didactic-mathematical knowledge on specific topics that can establish a line between pre-service and in-service teachers’ knowledge for teaching. Objective: The research aims to compare derivative knowledge of pre-service and in-service teachers to identify similarities and differences between teachers’ knowledge. Design: This research is a mixed and interpretative study. Settings and Participants: The participants were 22 pre-service teachers, and 11 in-service teachers enrolled in a pre-service teacher education programme and a master’s programme, respectively. Data collection and participants: Data were collected based on a questionnaire designed purposefully for the study. Results: The results show that pre-service teachers lack both epistemic derivative knowledge, while in-service teachers not only have this knowledge but relates it to its use in teaching. Pre-service teachers may not be making sense of the concept of derivative means, much less related to teaching. Conclusions: The insufficiencies found in pre-service teachers’ knowledge justify the pertinence to design specific formative cycles to develop prospective teachers’ epistemic facet of didactic-mathematical knowledge. It is recommended that both in-service and pre-service teachers discuss activities in which they can identify and reflect on possible mistakes and errors made by students. The development of these formative cycles should consider the complexity of the global meaning of the derivative

Instrumental Origin in the Learning Process : The wxMaxima and the Polynomial Function

RESUMEN: Esta investigación aborda el uso del software Maxima y su interfaz gráfica para Microsoft Windows, wxMaxima, en un curso de matemáticas operativas de primer semestre de Ingeniería Agropecuaria en la Universidad de Antioquia, Seccional Suroeste. Algunas características de la génesis instrumental, el enfoque ontosemiótico de la cognición y la instrucción, y la orquestación instrumental fueron usadas para estudiar la manera en el que un software es usado por los estudiantes para resolver problemas matemáticos relacionados con los polinomios. Los resultados muestran que los estudiantes pueden articular recursos provistos por el software y los adecúan para responder cuestiones matemáticas; sin embargo, algunos estudiantes se apegan a los procedimientos algorítmicos para validar sus respuestas. Se logra identificar, además, un nuevo componente en la génesis instrumental que hemos denominado “cooperación”, reconfigurando el proceso de génesis instrumental.ABSTARCT: This research deals with the use of Maxima software and its graphical interface for Microsoft Windows, wxMaxima, in a first-year basic mathematics course of Agricultural Engineering in the University of Antioquia, Southeast branch campus. Some features of the instrumental origin, the onto-semiotic approach of the cognition and the training, and the instrumental orchestration were used for studying the way in which a software program is used by the students in order to solve mathematical problems related to polynomials. The results show that the student can associate the resources provided by this software program and adapt them for answering mathematical questions; however, some students adhere to algorithmic procedures for validating their answers. A new component in the instrumental origin was also identified which we have called “cooperation” reconfiguring with this the process

The antiderivative understanding by students in the first university courses

ABSTARCT: In this article, we present the results of a questionnaire designed to evaluate college students’ understanding of the antiderivative. Specifically, by civil engineering students when answering the questionnaire’ tasks, in order to identify and characterize the meanings on the antiderivative that are mobilized by them. In order to analyse the answers given, we used some theoretical and methodological notions provided by the theoretical model known as the Onto-Semiotic Approach (OSA) of mathematics cognition and instruction. The results show knowledge of antiderivative by the Civil Engineering students. Furthermore, the comparison between the mathematical activity of students provides information that allows concluding that the meanings that they mobilized might be shared among their communities

Towards a methodology for the characterization of teachers´ Didactic-Mathematical knowledge

ABSTARCT: This research study aims at exploring the useof some dimensions and theoretical-methodological tools suggested by the modelof Didactic-Mathematical Knowledge (DMK) for the analysis,characterization and development of knowledge that teachers should have inorder to efficiently develop within their practice. For this purpose, weanalyzed the activity performed by five high school teachers, in relation to anactivity about patterns suggested in the framework of the Master of MathematicsEducation Program at University of Los Lagos, Chile. As a result of theanalysis, it becomes evident that teachers can indeed solve items related tothe common content knowledge, but have certain difficulties when they faceitems that aim at exploring other dimensions of their knowledge, for example,about extended content knowledge, of resources and means, or of the affectivestate of students

Objectification processes related to rate and whole-part using the measure : a case study with seven graders

RESUMEN: El propósito de este artículo es investigar procesos de objetivación relacionados con dos acepciones del número racional, a partir de la medición. El enfoque investigativo es cualitativo y considera el estudio de un caso, el cual se focaliza en la interacción social y en la mediación instrumental. Las actividades diseñadas favorecieron procesos de objetivación en el seno de prácticas sociales orientadas hacia la solución de una tarea matemática sobre medición. La mediación como un punto de partida permitió establecer un vínculo con la producción cultural histórica de los artefactos que usan los niños no únicamente para producir y reproducir conocimiento sino también para apropiarse de significados culturales.ABSTARCT: The objective of this paper is to research objectification processes related to the rational number as a ratio and as a whole number through measuring tasks. e research approach is qualitative and considers a case study which focuses not only on the social interaction but also on the instrumental mediation. e tasks were designed to promote the objectification processes of the rational number as a ratio and as a whole number within the social practice which fosters the mathematics task solution. Mediation as a starting point promoted linking the cultural production of artifacts used by kids not only to produce and reproduce knowledge but also to take over cultural meanings

A Proposal to Enhance Preservice Teacher’s Noticing

ABSTARCT: This paper presents a proposal for the development of ‘Teacher noticing’ with the help of dimensions taken from the teacher’s didactic-mathematical knowledge model, that makes up a Guide for Didactical Reflection. The proposal is based on a study conducted during a time span of two years while the preservice teachers were in their teaching practicum. The methodology was qualitative and data were taken in video, audio, observing rubrics, and interviews formats. The results not only indicate that the Guide can help develop teachers’ noticing, in order to identify and to question several specific domains of teacher knowledge involved in mathematics teaching, but also suggest that prospective teachers are moved into action and translate their noticing in didactic action during class. We hypothesize that preservice effective noticing occurs at the intersection of practice and discussion over the practice, with the help of ‘noticing tools’ with respect to the mathematical activities of students

Mathematics preservice teachers’ argumentation

ABSTARCT: This research deals with the preservice teachers’ dialogic argumentationswhen presenting geometry tasks to their colleagues, during discussion sessions previous to teaching to children. An argumentation analysis tool is used that complement Toulmin’s analysis proposal and that includes features related to mathematicallogic, rhetoric and dialectic features.We propose both a representation for the dialogic argumentation and a way to identify its structural qualitie