Towards a newer notion: noticing languages for mathematics content teaching

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 the articulated knowledge to the design of content-specific materials oriented towards enhancing the development of noticing processes with primary school student teachers in mathematics teacher training programmes. We propose processes of identification, interpretation and decision on languages for content teaching aimed at reducing school learning challenges, and developmental work at the levels of specialised word names and content-related explanatory and exemplifying sentences.PID2020-116514GB-I00 (Spanish Government), PID2019-104964GB-100 (Spanish Government), and Group GIPEAM 2017-SGR-101 (Catalan Government)

Raciocínios dos estudantes em tarefas de comparação, ordenação e representação de frações e números decimais

A cross-sectional study from 5th to 10th grade has been performed to analyse the levels of success and students’ reasoning in fraction and decimal number comparison tasks, and tasks of representing fractions and decimal numbers in the number line. Our study provides evidence of the use of different incorrect reasonings inferred in previous quantitative studies and also provides information on how these reasonings are used along grades. Results show that, although the natural number-based reasoning decreased, other incorrect reasoning appears in these types of tasks.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 decimales. Nuestro estudio aporta evidencias del uso de diferentes razonamientos incorrectos inferidos en estudios cuantitativos y, además, aporta información sobre su evolución. Los resultados muestran que, aunque disminuyó el razonamiento centrado en el uso del conocimiento del número natural, aparecen otros razonamientos incorrectos en este tipo de actividades.Realizou-se um estudo transversal desde o 5º ano do Ensino Básico ao 4º ano do Ensino Secundário, no qual se analisam os níveis de sucesso e raciocínios dos alunos em tarefas de comparação de frações, comparação de números decimais e representação de frações e decimais na reta numérica. Nosso estudo fornece evidências do uso de diferentes incorretos raciocínios inferidos em estudos quantitativos e fornece informações sobre como eles evoluem. Os resultados mostram que, embora o raciocínio focado no uso do conhecimento do número natural tenha diminuído, outros raciocínios incorretos aparecem nesse tipo de atividade

Raciocínios dos estudantes em tarefas de comparação, ordenação e representação de frações e números decimais

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 decimales. Nuestro estudio aporta evidencias del uso de diferentes razonamientos incorrectos inferidos en estudios cuantitativos y, además, aporta información sobre su evolución. Los resultados muestran que, aunque disminuyó el razonamiento centrado en el uso del conocimiento del número natural, aparecen otros razonamientos incorrectos en este tipo de actividades.A cross-sectional study from 5th to 10th grade has been performed to analyse the levels of success and students’ reasoning in fraction and decimal number comparison tasks, and tasks of representing fractions and decimal numbers in the number line. Our study provides evidence of the use of different incorrect reasonings inferred in previous quantitative studies and also provides information on how these reasonings are used along grades. Results show that, although the natural number-based reasoning decreased, other incorrect reasoning appears in these types of tasks.Realizou-se um estudo transversal desde o 5º ano do Ensino Básico ao 4º ano do Ensino Secundário, no qual se analisam os níveis de sucesso e raciocínios dos alunos em tarefas de comparação de frações, comparação de números decimais e representação de frações e decimais na reta numérica. Nosso estudo fornece evidências do uso de diferentes incorretos raciocínios inferidos em estudos quantitativos e fornece informações sobre como eles evoluem. Os resultados mostram que, embora o raciocínio focado no uso do conhecimento do número natural tenha diminuído, outros raciocínios incorretos aparecem nesse tipo de atividadeEsta investigación se ha llevado a cabo con el apoyo de la Conselleria d’Educació. Investigació, Cultura i Esport (Generalitat Valenciana, España) (PROMETEO/2017/135) y con el apoyo del Ministerio de Ciencia e Innovación (PID2020-116514GB-I00

Various ways to determine rational number size: an exploration across primary and secondary education

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 comparison tasks, recent research has shown that other incorrect strategies such as gap thinking or reverse bias can also explain these difficulties. The present study aims to investigate students’ different ways of thinking when working on fraction and decimal comparison tasks. The participants were 1,262 primary and secondary school students. A TwoStep Cluster Analysis revealed six different student profiles according to their way of thinking. Results showed that while students’ reasoning based on the properties of natural numbers decreased along primary and secondary school, almost disappearing at the end of secondary school, students’ reasoning based on gap thinking increased along these grades. This result seems to indicate that when students overcome their reliance on natural numbers, they enter a stage of qualitatively different errors before finally reaching the stage of correct understanding.This research was carried out with the support of Conselleria d’Educació, Investigació, Cultura i Esport (Generalitat Valenciana, Spain) (PROMETEO/2017/135) and with the support of the University of Alicante (UAFPU2018-035)

Incorrect Ways of Thinking About the Size of Fractions

The literature has amply shown that primary and secondary school students have difficulties in understanding rational number size. Many of these difficulties are explained by the natural number bias or the use of other incorrect reasoning such as gap thinking. However, in many studies, these types of reasoning have been inferred from comparing students’ accuracies in multiple-choice items. Evidence that supports that these incorrect ways of reasoning are indeed underlying is scarce. In the present work, we carried out interviews with 52 seventh grade students. The objective was to validate the existence of students’ incorrect ways of thinking about fraction size that were previously inferred from patterns of correct and incorrect answers to multiple-choice items, by looking at students’ verbalizations, and examine whether these ways of thinking are resistant to change. Students’ verbalizations support the existence of the different incorrect ways of thinking inferred from previous studies in fraction size. Furthermore, the high levels of confidence in their incorrect reasoning and the fact that they were reluctant to change their answer when they were confronted with other reasoning suggest that these ways of thinking may be resistant to change.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This research was carried out with the support of Conselleria d’Educació, Investigació, Cultura i Esport (Generalitat Valenciana, Spain) (PROMETEO/2017/135), the support of the postdoctoral grant (I-PI 69-20), and with the support of the Academy of Finland (Grant 336068, growing mind GM2, PI Minna Hannula-Sormunen)

Perfiles en la comprensión de la densidad de los números racionales en estudiantes de educación primaria y secundaria

The present cross-sectional study investigated 953 fifth to tenth grade students’ understanding 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 analysis highlighted qualitatively different ways of thinking: i) the idea of consecutiveness, ii) the idea of a finite number of numbers, and iii) the idea that between fractions, there are only fractions, and between decimals, there are only decimals. Furthermore, our profiles showed differences regarding rational number representation since students first recognised the dense nature of decimal numbers and then of fractions. Learners, however, were still found to have a natural number-based idea of the rational number structure by the end of secondary school, especially when they had to write a number between two pseudo-consecutive rational numbers.En este estudio transversal sobre la densidad de los números racionales participaron 953 estudiantes desde 5º curso de educación primaria hasta 4º curso de educación secundaria. Tras un análisis inductivo, codificando las respuestas a tres tipos de ítems, se llevó a cabo un análisis clúster, que reveló diferentes perfiles intermedios en la comprensión de la densidad. Se identificaron formas de pensar diferentes: i) la idea de consecutivo, ii) la idea de número finito de números, y iii) la idea de que entre fracciones solo hay fracciones y entre decimales solo hay decimales. Además, se obtuvieron diferencias con respecto a la representación de los números racionales: los estudiantes primero reconocieron la densidad en números decimales y posteriormente, en fracciones. Se destaca que los estudiantes al final de la educación secundaria todavía tenían una idea basada en el conocimiento del número natural, especialmente cuando tenían que escribir un número entre dos números racionales pseudo-consecutivos.This research was carried out with the support of Conselleria d’Educació, Investigació, Cultura i Esport (Generalitat Valenciana, Spain) (PROMETEO/2017/135), the support of the postdoctoral grant (I-PI 69-20), and with the support of the Academy of Finland (Grant 336068, growing mind GM2, PI Minna Hannula-Sormunen)

Natural number bias: a study of students’ reasoning in rational number multiplication

Una de las principales causas de las dificultades de los estudiantes de educación primaria y secundaria con las operaciones con los números racionales se debe al uso inapropiado de su conocimiento de los números naturales. Este fenómeno es conocido como natural number bias. Nuestra investigación tiene como objetivo examinar los niveles de éxito y los razonamientos de los estudiantes desde 5.º de educación primaria hasta 4.º de educación secundaria cuando resuelven tareas de multiplicación de un número natural por un racional. Los participantes fueron 438 estudiantes españoles de educación primaria y secundaria. Los resultados muestran porcentajes de éxito menores en tareas donde el conocimiento de los números naturales no es compatible para resolverlas. El análisis de los razonamientos de los estudiantes en estas tareas evidencia la existencia del fenómeno natural number bias en educación primaria y secundaria, pero mostrando su disminución en los últimos años. Estos resultados amplían y apoyan los resultados obtenidos en previas investigaciones cuantitativas.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 students’ reasoning from 5th to 10th grade when solving multiplication tasks of a natural number by a rational number. Participants were 438 Spanish primary and secondary school students. Results show lower percentages of success in tasks where knowledge of natural numbers was not compatible to solve them. The analysis of students’ reasoning in these tasks shows the existence of the natural number bias phenomenon in primary and secondary education, although it decreases during the last grades. These findings extend and support results about this phenomenon obtained in previous quantitative studies.Esta investigación se ha llevado a cabo con el apoyo de la Conselleria d’Educació, Investigació, Cultura i Esport (Generalitat Valenciana, España) (PROMETEO/2017/135) y con el apoyo de la Universidad de Alicante (UAFPU2018-035)

A competência olhar profissionalmente dos futuros professores de matemática: utilização de representações da prática

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 en los programas de formación de profesorado de educación infantil, primaria y secundaria. El Grupo de Investigación en Didáctica de las Matemáticas de la Universidad de Alicante (GIDIMAT-UA) ha contribuido a esta agenda de investigación y ha aportado características del desarrollo de esta competencia y de los entornos de aprendizaje que apoyan su desarrollo. Una de las características de los entornos de aprendizaje es el uso de representaciones de la práctica (viñetas) que proporcionan contextos reales para interpretar aspectos de la enseñanza y aprendizaje de las matemáticas y, por consiguiente, proporcionan oportunidades para relacionar las ideas teóricas con ejemplos de la práctica. Se presentan ejemplos de viñetas que forman parte de los entornos de aprendizaje de los programas de formación de profesores y que tienen el objetivo de desarrollar esta competencia. Además, se presentan algunos resultados que muestran características de cómo se desarrolla esta competencia.Professional noticing of mathematical teaching-learning situations implies that teachers are able to identify relevant aspects of a situation, use their knowledge to interpret them and decide how to continue the instruction. This competence has been identified as one of the important competences to be developed in teacher training programmes for early childhood, primary and secondary education. The Research Group in Didactics of Mathematics at the University of Alicante (GIDIMAT-UA) has contributed to this research agenda by providing characteristics of how this competence is developed and characteristics of the learning environments that support its development. One of the features of learning environments is the use of representations of practice (vignettes) that provide real contexts for interpreting aspects of mathematics teaching and learning and thus provide opportunities to relate theoretical ideas to examples from practice. Examples of vignettes that are part of learning environments in teacher education programmes aiming to develop this competence are presented. In addition, some results are presented showing characteristics of how this competence is developed.Olhar profissionalmente para situações de ensino e de aprendizagem matemática implica que os professores são capazes de identificar aspectos relevantes de uma situação, usar seus conhecimentos para interpretá-los e decidir como continuar ensinando. Esta competência foi identificada como uma das competências importantes a serem desenvolvidas nos programas de formação de professores para a primeira infância, ensino primário e secundário. O Grupo de Pesquisa em Didática da Matemática da Universidade de Alicante (GIDIMAT-UA) contribuiu para esta agenda de pesquisa ao fornecer características de como esta competência é desenvolvida e características dos ambientes de aprendizagem que apoiam seu desenvolvimento. Uma das características dos ambientes de aprendizagem é o uso de representações da prática (vinhetas) que fornecem contextos reais para interpretar aspectos do ensino e aprendizagem da matemática e, assim, oferecem oportunidades para relacionar ideias teóricas a exemplos da prática. São apresentados exemplos de vinhetas que fazem parte de ambientes de aprendizagem em programas de formação inicial de professores e que visam desenvolver esta competência. Além disso, são apresentados alguns resultados que mostram características de como essa competência é desenvolvida.Esta investigación ha recibido el apoyo del proyecto PID2020-116514GB-I00 financiado por el Ministerio de Ciencia e Innovación, España y el apoyo del Programa de la Unión Europea Erasmus+ (project coReflect@maths, 2019–1–DE01–KA203–004947)

Profiles in understanding the density of rational numbers among primary and secondary school students Perfiles en la comprensión de la densidad de los números racionales en estudiantes de educación primaria y secundaria

The present cross-sectional study investigated 953 fifth to tenth grade students' understanding 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 analysis highlighted qualitatively different ways of thinking: i) the idea of consecutiveness, ii) the idea of a finite number of numbers, and iii) the idea that between fractions, there are only fractions, and between decimals, there are only decimals. Furthermore, our profiles showed differences regarding rational number representation since students first recognised the dense nature of decimal numbers and then of fractions. Learners, however, were still found to have a natural number-based idea of the rational number structure by the end of secondary school, especially when they had to write a number between two pseudo-consecutive rational numbers.En este estudio transversal sobre la densidad de los números racionales participaron 953 es-tudiantes desde 5º curso de educación primaria hasta 4º curso de educación secundaria. Tras un análisis inductivo, codificando las respuestas a tres tipos de ítems, se llevó a cabo un análisis clúster, que reveló diferentes perfiles intermedios en la comprensión de la densidad. Se identificaron formas de pensar dife-rentes: i) la idea de consecutivo, ii) la idea de número finito de números, y iii) la idea de que entre fracciones solo hay fracciones y entre decimales solo hay decimales. Además, se obtuvieron diferencias con respecto a la representación de los números racionales: los estudiantes primero reconocieron la densidad en núme-ros decimales y posteriormente, en fracciones. Se destaca que los estudiantes al final de la educación se-cundaria todavía tenían una idea basada en el conocimiento del número natural, especialmente cuando tenían que escribir un número entre dos números racionales pseudo-consecutivos.</p

Influence of clinical and neurocognitive factors in psychosocial functioning after a first episode non-affective psychosis: differences between males and females

BackgroundDeficits in psychosocial functioning are present in the early stages of psychosis. Several factors, such as premorbid adjustment, neurocognitive performance, and cognitive reserve (CR), potentially influence functionality. Sex differences are observed in individuals with psychosis in multiple domains. Nonetheless, few studies have explored the predictive factors of poor functioning according to sex in first-episode psychosis (FEP). This study aimed to explore sex differences, examine changes, and identify predictors of functioning according to sex after onset.Materials and methodsThe initial sample comprised 588 individuals. However, only adults with non-affective FEP (n = 247, 161 males and 86 females) and healthy controls (n = 224, 142 males and 82 females) were included. A comprehensive assessment including functional, neuropsychological, and clinical scales was performed at baseline and at 2-year follow-up. A linear regression model was used to determine the predictors of functioning at 2-year follow-up.ResultsFEP improved their functionality at follow-up (67.4% of both males and females). In males, longer duration of untreated psychosis (β = 0.328, p = 0.003) and worse premorbid adjustment (β = 0.256, p = 0.023) were associated with impaired functioning at 2-year follow-up, while in females processing speed (β = 0.403, p = 0.003), executive function (β = 0.299, p = 0.020) and CR (β = −0.307, p = 0.012) were significantly associated with functioning.ConclusionOur data indicate that predictors of functioning at 2-year follow-up in the FEP group differ according to sex. Therefore, treatment and preventative efforts may be adjusted taking sex into account. Males may benefit from functional remediation at early stages. Conversely, in females, early interventions centered on CR enhancement and cognitive rehabilitation may be recommended