Article

An Empirical Classification Model for Errors in High School Mathematics

Abstract

Six categories of errors were found inclusive and mutually exclusive in a systematic conten-analysis of students' errors in high school mathematics graduation exams
... The authors [35] consider the research of errors as necessary, even to find out if a certain teaching style is associated to certain errors, in particular. For the purposes of the theoretical framework of this research, it has been considered the category proposed by these authors which is proposed hereafter: 1) Errors due to misused data: It includes errors that may be related to a disagreement between data from the problem and how the student processed it. ...
... The students' errors have been identified, and the analyzed data has been grouped according to the [35] classification of errors, and the problems according to the type of mathematical competence by [27]. The following results have been obtained. ...
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In this article, the performance and errors are featured and analysed in the limit problems solving of a real-valued function, in correspondence to competency-based education in engineering careers, in the south of Chile. The methodological component is contextualised in a qualitative research, with a descriptive and explorative design, with elaboration, content validation and application of quantitative instruments, consisting of two parallel forms of open answer tests, based on limit application problems. The mathematical competences and errors made by students from five engineering careers from a public University are identified and characterized. Results show better performance only to solve routine-context problem-solving competence, thus they are oriented towards a rational solution or they use a suitable problem-solving method, achieving the correct solution. Regarding errors, most of them are related to techniques and the incorrect use of theorems and definitions of real-valued function limits of real variable.
... The repetitive nature of feedback is a main problem, as most student answers contain systematic error patterns, meaning that different students often make similar mistakes [32,43]. In paper-based assessment processes (PP), teachers try to provide targeted, relevant, original, and personal feedback. ...
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Feedback has been recognized as a crucial element in the learning and teaching process. Although teachers know and accept this, they are not always eager to engage in this tedious and time-consuming activity. This study investigates how computers can work together with teachers to make the process of giving feedback more efficient by introducing a semi-automated approach (SA) with reusable feedback: when a teacher writes feedback for a student, the computer saves it, so it can be reused when following students make similar mistakes. We devised the concept of atomic feedback, a set of form requirements that could enhance feedback's reusability. To write atomic feedback, teachers have to identify the independent errors and write brief feedback items for each separate error. Our SA approach with reusable feedback was implemented in Moodle. During a crossover experiment with math teachers (n = 36 + 9 in pilot study), we examined (1) whether SA saves time or changes the amount of feedback, as compared to traditional, paper-based correction work, (2) the extent to which the feedback was atomic, (3) whether atomic feedback enhances the reusability of feedback and (4) how teachers used and perceived the SA system. In light of the results, which suggest that atomic feedback is indeed reusable, we propose formal requirements for writing reusable feedback. Nevertheless, teachers did not save time using the SA system, but they provided significantly more feedback.
... The error in the mathematical field is considered a natural event, common in the learner's trajectory, regardless of age and/or performance level [13]. Models or theories about classification of mathematical errors are presented in studies such as those of [13], [18], [22], [24] and [27]. ...
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Intelligent Tutoring Systems (ITS) environments have the ability to adapt to each learner’s individual needs and thus provide immediate and personalized instructions, both in content and in form. This personalization can consider several aspects, such as the interaction, the level of knowledge, the error, and the affective state of the learner aiming to improve the teaching strategies. One of the strategies is the possibility of presenting tutorial interventions when verifying an error made in solving an exercise, or when detecting that the learner is unmotivated or frustrated. These tutorial interventions can improve teaching methods in order to improve performance and the level of knowledge acquired by the learner. In this sense, this research presents a model that allows the automatic presentation of tutoring interventions based on identification of mathematical error kind committed by the learner, in addition to inferring his affective state. Experiments were carried out in a real learning environment, using the proposed model implemented in a fraction game. In general, the results presented indicate that personalized tutorial interventions favor greater engagement and motivation of learners and improvement in learning outcomes.
... Among the current research lines is the diagnosis, analysis, and documentation of the most common errors, and their classification (Abrate, Pochulu & Vargas, 2006;Movshovitz-Hadar, Inbar, & Zaslavksy, 1987;Radatz, 1980). We base this work on the classification proposed by Abrate et al. (2006), supported by the studies of Movshovitz-Hadar et al. (1987) and Radatz (1980), which contemplates the following categories: ...
... Radatz [15] offers a systematic categorization of errors built around the difficulties arising from information processing. His influential classification has been adopted and modified in subsequent years by authors like References [16][17][18][19][20], among many others. Three major categories of errors are usually identified: (i) those referred to as the calculations themselves, (ii) those having to do with the technical description (and the use of the language specific to mathematics), and (iii) those involving decision-making strategies. ...
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Although Error Analysis (EA) has been broadly used in Foreign Language and Mother Tongue learning contexts, it has not been applied in the field of engineering and by STEM (Science, Technology, Engineering, and Mathematics) students in a systematic way. In this interdisciplinary pilot study, we applied the EA methodology to a wide corpus of exercises and essays written by third-year students of mechanical engineering, with the main purpose of achieving a precise diagnosis of the students’ strengths and weaknesses in writing skills. For the analysis to be as exhaustive as possible, the errors were typologized into three main categories (linguistic, mathematical, and rhetorical–organizational), each of which is, in turn, subdivided into 15 items. The results show that the predominant errors are rhetorical–organizational (39%) and linguistic (38%). The application of EA permits the precise identification of the areas of improvement and the subsequent implementation of an educational design that allows STEM students to improve their communicative strategies, especially those related to the writing skills and, more precisely, those having to do with the optimal use of syntax, punctuation, rhetorical structure of the text, and mathematical coherence.
... Entre los estudios relativos a geometría en Franchi y Hernández [10] se utiliza la observación participante para hacer un compendio de los errores cometidos a lo largo de dos semestres por alumnos de Geometría de la Universidad de Zulia. Posteriormente se clasificaron dichos errores atendiendo a las categorizaciones de Brousseau [6], Socas [23], Movshovitz y otros [15], Radatz [16] y Astolfi [2]. Se cuantificó el número de errores de los tipos propuestos por cada uno de los autores, así como los errores que no era posible identificar con ninguno de los tipos. ...
... Entre los estudios relativos a geometría en Franchi y Hernández [10] se utiliza la observación participante para hacer un compendio de los errores cometidos a lo largo de dos semestres por alumnos de Geometría de la Universidad de Zulia. Posteriormente se clasificaron dichos errores atendiendo a las categorizaciones de Brousseau [6], Socas [23], Movshovitz y otros [15], Radatz [16] y Astolfi [2]. Se cuantificó el número de errores de los tipos propuestos por cada uno de los autores, así como los errores que no era posible identificar con ninguno de los tipos. ...
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Nos planteamos la siguiente cuestión: ¿cuáles son los errores y dificultades que aparecen más habitualmente en el primer contacto del alumnado con el concepto de vector? Se pretende registrar y clasificar los errores y dificultades de aprendizaje del alumnado de 3º de ESO con los contenidos relativos a vectores en el plano. A partir dela experiencia realizada en un centro de secundaria en el curso 2012-2013 se concreta lo anterior en los siguientes objetivos: determinar los errores que el profesorado con experiencia en la materia identifica como más habituales en relación con las causas de los mismos, relacionar los errores encontrados con las deficiencias en objetivos de aprendizaje que el alumnado debería haber adquirido previamente y proponer cambios que introduzcan mejoras en este sentido
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RESUMEN La capacidad de aplicar las matemáticas en una situación del mundo real se considera un objetivo principal de las matemáticas y de la formación de ingenieros. El propósito de este estudio, es identificar y caracterizar las competencias matemáticas y los errores de los estudiantes de ingenierías en la resolución de problemas de límites de funciones reales, a través de la aplicación de instrumentos evaluativos con problemas de respuesta abierta. Se presenta la metodología cuantitativa del estudio descriptivo, con una muestra de cinco carreras de ingeniería de la Universidad de Los lagos en sus campus de Puerto Montt y Osorno. Considerando las etapas de resultado y completación, donde el problema está casi resuelto o se utilizó un método apropiado que permitió llegar a la solución correcta, los resultados indican que los estudiantes de las cinco ingenierías registraron desempeños similares, mostrando un mejor resultado en la competencia tipo 2 de problemas rutinarios de contexto realista y fantasista, pero mayoritariamente con errores de uso de teoremas y definiciones deformadas y errores técnicos. Palabras clave: competencias matemáticas, resolución de problemas, límite, errores, ingenierías.
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This research focuses on the current state of the Civil Engineering program in Colombia, a developing country, which to date has not fully embraced a process of unification of content at the country level, such as the one observed successful way in the Bologna process or in the Tuning project. The analyzed information is available in the National System of Higher Education, institutional repository of the Ministry of Higher Education of Colombia, and categorized by geographical regions, how is this country geopolitically organized, attending to the attributes: character of the institution of higher education (IES), number of credits and duration. Likewise, emphasis is placed on the importance of the approximation of all HEIs towards quality programs that unify generic and specific competences, thus achieving a closer approximation to what was expected with the Tuning project, as occurred in Europe, bearing in mind that The progress of society is achieved to the extent that you have a quality education.
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