Ceneida Fernández

University of Alicante | UA · Departamento de Innovación y Formación Didáctica

Research has shown that there is a need to examine prospective teachers’ development trajectories related to noticing expertise. An important content in the Spanish high school curriculum (16–18 years old) is the limit concept. Given the importance of this concept in the curriculum and the difficulties some prospective teachers have, developing the...

Se ha llevado a cabo un estudio transversal desde 5º de Educación Primaria hasta 4º de Educación Secundaria (ESO), en el que se analiza los niveles de éxito y razonamientos de los estudiantes en tareas de comparación de fracciones, comparación y ordenación de números decimales, y de representación en la recta numérica de fracciones y números decima...

Rational number density has been investigated through open-ended question tasks and multiple-choice tasks or by asking to interpolate a number between two numbers. However, students' responses to these three types of tasks were not directly compared. The objective is to look for relationships between the three types of tasks in order to identify di...

Investigaciones previas han mostrado que la competencia docente mirar profesionalmente la enseñanza de las matemáticas se puede desarrollar en los programas de formación inicial a través de entornos de aprendizaje que permitan desarrollar las destrezas de reconocer e interpretar aspectos relevantes de la enseñanza de las matemáticas y decidir cómo...

En el estudio se analiza la forma en que los estudiantes de Primaria y Secundaria resuelven problemas de estructura multiplicativa (multiplicación, división-partitiva y división-medida). Se utilizó un cuestionario con nueve problemas en los que se tomó en cuenta el uso de números naturales y fracciones, y se analizó tanto el nivel de éxito como las...

Mirar profesionalmente las situaciones de enseñanza-aprendizaje de las matemáticas implica que el profesorado sea capaz de identificar aspectos relevantes en una situación, usar su conocimiento para interpretarlos y decidir cómo continuar en la enseñanza. Esta competencia ha sido identificada como una de las competencias importantes a desarrollar e...

The present cross-sectional study investigated 953 fifth to tenth grade students’ understand-ing of the dense structure of rational numbers. After an inductive analysis, coding the answers based on three types of items on density, a TwoStep Cluster Analysis revealed different intermediate profiles in the understanding of density along grades. The a...

In this report, we revise and connect our approaches to mathematics teacher noticing and to the classroom language of the teacher for content teaching in the attempt: 1) to articulate mathematics teacher education knowledge from research on noticing and on language around a newer notion of noticing languages for content teaching; and 2) to apply th...

Problem solving is a core mathematical activity and part of curricula and national standards worldwide. Studies show that the flexible use of strategies (e.g., adapting the strategy to a given task) leads to higher performance in problem solving but also that many children struggle with applying strategies in a flexible way (Elia et al., 2009). The...

Learning support is a key aspect of the teaching profession. In particular, promoting mathematics-related learning is a goal when mathematics teachers respond to students' questions or comments during their work on tasks. "Helping" learners in this sense should be (A) adaptive to the learner's needs and (B) carry the potential to elicit further lea...

This paper is part of a larger study focuses on a teaching experiment (pre-test, instruction, post-test) that aims to analyse primary school students' features on the transition from natural to rational numbers when solving multiplicative structure problems. Here, we analysed 61 6th graders responses to nine multiplicative structure problems with n...

This volume contains the papers presented at the International Conference Building on the Past to Prepare for the Future held from August 8-13, 2022, in King’s College, Cambridge, UK. It was the 16th conference organised by The Mathematics Education for the Future Project - an international educational and philanthropic project founded in 1986 and...

During the last decades, research in teacher noticing has increased since its development is considered important in teacher training programs. An issue that needs more research is the relationship between teachers’ mathematical knowledge for teaching in a specific mathematical domain and their ability to notice. This study focuses on how pre-servi...

Este estudio tiene como objetivo analizar cómo estudiantes españoles de 3º a 6º grado (8-12 años) resuelven problemas realistas, en particular, de división-medida y división-partitiva con resto. Los participantes resolvieron seis problemas y el análisis se centró en el nivel de éxito y las estrategias que utilizaron a lo largo de los grados. Los re...

The density of rational numbers is considered the most difficult domain in which the natural number bias persists. The objective of this study is to identify primary and secondary school students’ ways of answering about density. We extent previous research by conducting a cross-sectional study from 5th grade of primary school to 4th grade of secon...

This research investigated students' intermediate stages in understanding the dense structure of rational numbers. A cross-sectional study with 953 students from 5 th to 10 th grade was performed. After an inductive analysis coding the open answers to a question about how many numbers there are between two given rational numbers, a TwoStep Cluster...

This poster conveys how vignette activities can be used to promote critical reflection by pre-service teachers about aspects of teaching during their training. It also demonstrates one way to construct vignettes as a method to elicit beliefs, values and understanding for research purposes.

Students often show difficulties in understanding rational numbers. Often, these are related to the natural number bias, that is, the tendency to apply the properties of natural numbers to rational number tasks. Although this phenomenon has received a lot of research interest over the last two decades, research on the existence of qualitatively dif...

Decidir cómo continuar la enseñanza se ha identificado como la destreza más difícil de entre las tres que configuran la competencia de mirar profesionalmente el pensamiento matemático del estudiante. En este estudio 95 estudiantes para maestro de Educación Primaria resolvieron una tarea en la que debían proponer un objetivo de aprendizaje y activid...

Many studies have addressed the natural number bias in fraction comparison, focusing on the role of congruency. However, the congruency effect has been observed to operate in the opposite direction, suggesting that a deeper explanation must underlie students' different reasoning. W e extend previous research by examining students' reasoning and by...

The goal of this study is to characterize how pre-service teachers recognize students’ reasoning, based in the unitizing process, when students solve ratio comparison problems and, how they consider the unitizing process as a Key Development Understanding to propose teaching decisions. Ninety-one pre-service primary teachers interpreted primary sch...

One of the main causes of primary and secondary school students’ difficulties with the rational numbers operations is the inappropriate use of their natural number knowledge. This phenomenon is known as natural number bias. Our research analyses students’ success levels and El fenómeno natural number bias… 33 Quadrante, Vol. 28, N.º 2, 2019 student...

Understanding rational numbers is a complex task for primary and secondary school students. Previous research has shown that a possible reason is students’ tendency to apply the properties of natural numbers (inappropriately) when they are working with rational numbers (a phenomenon called natural number bias). Focusing on rational number compariso...

The objective of this study was to identify cognitive apprehensions used by fifth- and sixth-grade students (10–12-year-olds) when answering far generalization questions in two problems of visual pattern generalization. A total of 81 students solved two linear generalizing problems, presented in two different configurations, in a succession of figu...

Previous quantitative studies have shown students' difficulties in understanding the magnitude of fractions. This is often explained by the natural number bias or the use of other incorrect reasoning such as gap thinking. However, qualitative studies that support these interpretations are scarce. In the current research, we interview 52 seventh gra...

Professional noticing is understood as a knowledge-based reasoning process. Particularly, noticing students' mathematical understanding involves the use of different domains of knowledge to attend to students' strategies, to interpret students' understanding and to decide how to respond considering students' understanding. In this paper, we discuss...

Developing teachers' knowledge, awareness and noticing ability in and for attending and developing students' mathematical understanding needs to be a core element in teacher education-mainly taking as a starting point the students' reasoning. Pursuing such gold one possible perspective is to consider (pre-service) teacher's narratives on observed p...

Numerous quantitative studies have shown that learners struggle to understand the magnitude of fractions. This is often explained by the natural number bias or the use of gap thinking. However, qualitative evidence that supports the interpretations of this quantitative research is scarce. We have carried out a qualitative study interviewing 52 seve...

The objective of this research is to understand how a learning trajectory supports pre-service primary teachers in their noticing of students’ mathematical understanding. A total of 95 pre-service primary school teachers used a learning trajectory related to the part-whole meaning of the fraction concept to interpret students’ understanding and pro...

This study explores relationships among prospective secondary teachers’ skills of attending to relevant mathematics elements in students’ answers, interpreting students’ mathematical understanding, and proposing instructional actions. Thirty prospective secondary mathematics teachers analyzed three high school students’ answers to three problems of...

Previous research has shown evidence of pre-service teachers’ development of the skill of noticing students’ mathematical understanding when they participate in online debates, throughout meetings with colleagues or watching video clips. The aim of our study is to analyze if writing narratives and receiving feedback from a university tutor could he...

Los números racionales es uno de los contenidos matemáticos más difíciles de comprender por los estudiantes de Educación Primaria y Secundaria, ya que a menudo aplican las propiedades de los números naturales para resolver actividades con los números racionales cuando no es apropiado (fenómeno Natural Number Bias). Nuestra investigación examina la...

En este estudio 85 estudiantes para maestro participaron en un módulo de enseñanza donde tenían que usar la información de una trayectoria hipotética de aprendizaje de estudiantes de educación primaria sobre las fracciones para interpretar el pensamiento matemático de los estudiantes y decidir cómo responder. Los resultados muestran que la informac...

Since noticing has been identified as a critical component of teaching expertise, researchers have tried to identify contexts to its development. These studies assume that growth in teachers' noticing expertise can be inferred from their professional discourse. Prior research has also shown that teachers' noticing development in teacher education p...

The aim of this paper is to examine whether the use of a hypothetical learning trajectory as a guide to notice students' mathematical thinking could improve pre-service teachers professional discourse and enhance pre-service teachers' noticing. Twenty-nine pre-service primary school teachers participated in a learning environment in which they had...

Natural number bias in fraction comparison has been studied examining the role of congruency effect. However, the congruency effect has been observed in opposite directions , suggesting that an explanation may lie in the strategies used by students. One of the strategies that may be used by students is gap thinking. We have carried out a cross-sect...

We summarize results obtained by the Didactics of Mathematics research group at the University of Alicante on the competence of professional noticing. The research focused on three issues over the last years: (i) characterizing how the skills that make up professional noticing interrelate; (ii) characterizing different degrees of competence develop...

This article examines the characteristics of pre-service teachers' knowledge of subconstructs implied in proportional reasoning. The participants were 91 students at Universidad de Alicante (Spain), who solved twelve problems related to these subconstructs. A cluster analysis identified four partially nested profiles, in terms of the way the studen...

This article examines the characteristics of pre-service teachers’ knowledge of sub-constructs implied in proportional reasoning. The participants were 91 students at Universidad de Alicante (Spain), who solved twelve problems related to these sub-constructs. A cluster analysis identified four partially nested profiles, in terms of the way the stud...

The aim of this study is to examine to what extent the identification of the coordination of approximations in the domain and in the range is considered a Key Developmental Understanding (KDU in Simon, 2006) by prospective teachers. 25 prospective secondary school mathematics teachers anticipated high school students' answers to problems about the...

La competencia mirar profesionalmente se ha identificado como una competencia necesaria que todos los maestros deben adquirir como medio para identificar situaciones relevantes de aprendizaje en el aula. La conceptualización de esta competencia se ha abordado desde diferentes perspectivas, aunque desde todas ellas se subraya la importancia de iden...

Resumen. Mirar profesionalmente es una de las competencias que un maestro debe desarrollar. Las investigaciones en formación de maestros han mostrado que esta competencia puede empezar a desarrollarse en los programas de formación inicial. En este estudio, analizamos las respuestas de 31 estudiantes para maestro a dos tareas en las que debían inter...

Resumen. Este estudio examina el conocimiento del razonamiento proporcional de estudiantes para maestro (EPM) y cómo reconocen características de la comprensión de los estudiantes. 92 EPM resolvieron 12 problemas vinculados a 12 sub-constructos del razonamiento proporcional y analizaron tres res-puestas de estudiantes a cada uno de los problemas an...

Learning Trajectories are seen as a tool that can help pre-service teachers (PTs) focus on students’ mathematical thinking to make instructional decisions. The aim of this study is to examine PTs’ use of a Learning Trajectory for fractions to notice students’ fractional reasoning. Our results show that the use of a Learning Trajectory for fraction...

Este trabajo caracteriza la evolución del pensamiento algebraico en alumnos de Educación Primaria (6 a 12 años) cuando resuelven un problema de generalización lineal con una sucesión de figuras. Los resultados indican que el pensamiento algebraico se inicia con la coordinación de la estructura espacial y numérica puesta de manifiesto por el uso de...

This study investigates Spanish and Chinese students’ performance on both addition problems and proportion problems considering a cross-national perspective. The effect of number structure and nature of quantities was also considered. Nine hundred twenty-five 4th to 8th graders (453 Chinese, 472 Spanish) took a test which is composed of addition an...

Este estudio tiene como objetivo generar descriptores del desarrollo de la competencia docente "mirar profesionalmente" las situaciones de enseñanza-aprendizaje de las matemáticas. Analizamos las narrativas escritas por 39 estudiantes para maestro de educación primaria durante su periodo de prácticas en las que se les solicitaba identificar una sit...

This study focuses on how prospective teachers learn about students' mathematical thinking when (i) anticipating secondary students' answers reflecting different characteristics of understanding and (ii) propose new activities in relation to the classification of quadrilaterals. The data were collected from forty-eight prospective secondary school...

El objetivo de este estudio es caracterizar la evolución de los niveles de éxito y de las estrategias empleadas por estudiantes de Educación Primaria (edad 6-12 años) cuando resuelven problemas de estructura multiplicativa. Los resultados muestran que los estudiantes de 6-8 años emplearon mayoritariamente estrategias de modelización y conteo, pero...

The aim of this study is to characterize the evolution of the levels of success and strategies used by primary school students (6-12 years old) when solving multiplicative structure problems. Results show that students between 6-8 years old used mostly modelling and counting strategies, but after the third course, the majority of students used the...

Previous studies have focused on examining contexts and paths for the development of the skill of noticing children's mathematical thinking. However, the development of this skill is not an easy task and deserves further research. We analyze how the task of writing narratives could allow pre-service teachers to develop this skill. During their prac...

This research study examines the development of the ability of pre-service teachers to notice signs of students’ understanding of the derivative concept. It analyses preservice teachers’ interpretations of written solutions to problems involving the derivative concept before and after participating in a teacher training module. The results indicate...

Este estudio se centra en examinar la evolución en los niveles de éxito en la resolución de problemas de estructura multiplicativa por estudiantes de Educación Primaria (desde 1º a 6º curso, alumnos de 6 a 12 años). Los resultados indican que, en función de las categorías, los problemas de producto de medida fueron los más difíciles en todos los cu...

Resumen. El objetivo de este estudio es aportar información sobre el papel que desempeña el conocimiento de matemáticas de los estudiantes para maestro (EPM) cuando piensan en el aprendizaje de las matemáticas de los estudiantes de primaria. Nuestro estudio se centra en el razonamiento up and down que es una de las componentes que facilitan el desa...

Resumen Este estudio se centra en examinar el papel del conocimiento de matemáticas de los estudiantes para maestro (EPM) cuando piensan en el aprendizaje de la idea de razón en estudiantes de educación primaria. 92 EPM resolvieron una tarea formada por problemas y 3 respuestas a cada problema de estudiantes de educación primaria que mostraban dife...

El objetivo de esta investigacion es indagar en el conocimiento especializado de contenido matematico de estudiantes para maestro (EPM) sobre problemas de division-medida. Los EPM respondieron a dos cuestionarios en los que resolvieron dos problemas de division-medida e interpretaron respuestas dadas por alumnos de primaria. Un alto porcentaje de E...

One of the core ways allowing obtain an ampler and deeper understanding on the content of teachers' knowledge concerns analysing teachers' practices. Such practice can be perceived in a broader way, not limited to classroom practice. On the other side, discussing and reflecting on the same situation with different theoretical and methodological app...

The aim of this research is to understand the way in which students struggle with the distinction between dimensionality and “directionality” and if this type of potential confusion could be a factor affecting students’ tendency toward improper linear reasoning in the context of the relations between length and area of geometrical figures. 131 9th...

Un dominio particular del conocimiento matemático para la enseñanza es el conocimiento de matemáticas especializado. Este estudio se centra en examinar el conocimiento de matemáticas especializado en el ámbito del razonamiento proporcional de un grupo de estudiantes para maestro de Educación Primaria. Los resultados muestran que los estudiantes par...

Professional noticing of students’ mathematical thinking in problem solving involves the identification of noteworthy mathematical ideas of students’ mathematical thinking and its interpretation to make decisions in the teaching of mathematics. The goal of this study is to begin to characterize pre-service primary school teachers’ noticing of stude...

The goal of this research is to characterize prospective mathematics teachers’ development of professional noticing of students’ mathematical thinking in on-line contexts. Specifically, we are interested in how the participation in on-line discussions, when prospective teachers solve specific tasks, supports the development of professional noticing...

This study investigates the development of proportional and additive methods along primary and secondary school. In particular, it simultaneously investigates the use of additive methods in proportional word problems and the use of proportional methods in additive word problems. We have also studied the role played by integer and non-integer relati...

RESUMEN Este estudio analiza las relaciones implicativas entre las estrategias usadas por 136 estudiantes de primer curso de educación secundaria en la resolución de problemas lineales y no lineales. En primer lugar, se describen las estrategias ocupadas por los alumnos y después, empleando el software CHIC, se identifican sus relaciones implicativ...

This study analyzes the implicative relations between strategies used by 136 first year high school students in solving linear and non-linear problems. First of all, it describes the strategies used by students and then, using the CHIC software, it identifies implicative relations between them. The results demonstrate the importance of understandin...

The aim of this study is to determine the variation of students' behavioral profiles along Primary and Secondary School when solving proportional and non-proportional problems. The participants were 755 primary and secondary school students. From the analysis of students' answers we identify five profiles indicating that primary school students use...

Resumen. El objetivo de este estudio es determinar perfiles de comportamiento de los estudiantes cuando resuelven problemas proporcionales y no proporcionales y su variación a lo largo de la Educación Primaria y ESO. Los participantes del estudio fueron 755 estudiantes de Educación Primaria y ESO. El análisis de las respuestas nos permitió identifi...

Resumen Este estudio analiza la influencia de las relaciones multiplicativas entre las cantidades y su naturaleza discreta o continua en el desarrollo de la capacidad de los estudiantes de educación primaria de diferenciar situaciones aditivas y proporcionales. Los resultados han mostrado que la relación multiplicativa entre las cantidades influye...

The aim of this research is to identify aspects that support the development of prospective mathematics teachers' professional noticing in a b-learning context. The study presented here investigates the extent to which prospective secondary mathematics teachers attend and interpret secondary school students' proportional reasoning and decide how to...

This study investigates the effect of the number structure (integer vs. non-integer ratios) and the nature of the quantities (discrete vs. continuous) on the performance of secondary school students and on the strategies they use when solving proportional and non-proportional word problems. 551 secondary school students from 1st to 4th grade solved...

The purpose of this study is to examine the implicative relationship among students' strategies while solving proportional and nonproportional problems. We used the computer software CHIC to carry out an implicative statistical analysis of the strategies used in different types of problems. Our analysis showed that the use of some strategies was li...

The present study explores relationships between additive and multiplicative structures in the context of proportional reasoning. One goal is to examine hypothetical learning trajectories derived from these relations in the context of understanding proportionality. 198 Spanish primary school children were given a test which involved twelve problems...

This study explores the development of students' strategies from primary to secondary school when solving proportional and additive problems. Its goal is to identify characteristics of the development of proportional reasoning and how the use of integer and non-integer ratios and the discrete or continuous nature of quantities influence this develo...

Resumen Presentamos los resultados de un estudio sobre la evolución de los perfiles de los estu-diantes cuando resuelven problemas lineales y no lineales desde los 9 a los 16 años. Los resultados indican que hay una evolución en muchos de los estudiantes desde la utilización de las relaciones aditivas "en cualquier problema" durante la educación pr...

Universidad de Alicante RESUMEN El trabajo de investigación que se presenta aporta información para la selección y evaluación de recursos multimedia para el aprendizaje de conocimiento didáctico profesional, relativo a la enseñanza y aprendizaje de la geometría en educación infantil y primeros años de primaria. Siguiendo con el objetivo general doc...