Do students confuse dimensionality and “directionality”?

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 grade students were confronted with a multiple-choice test consisting of six problems related to the perimeter or the area of an enlarged geometrical figure, then some interviews were carried out to obtain qualitative data in relation to students’ reasoning. Results indicate that more than one fifth of the students’ answers could be characterized as based on directional thinking, suggesting that students struggled with the distinction between dimensionality and “directionality”. A single arrow showing one direction (image provided to the students) seemed to strengthen the tendency toward improper linear reasoning for the area problems. Two arrows showing two directions helped students to see a quadratic relation for the area problems.The research reported here has been financed in part by Ministerio de Educación y Ciencia, Dirección General de Investigación, Spain, under Grant no. EDU2011-27288 and in part by the University of Alicante under the birth project GRE10-10

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)

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)

A stratégiahasználat rugalmassága az alsó tagozatos matematikában: elemzés és fejlesztés

A neveléslélektan kutatói és a matematikát tanítók régóta hangsúlyozzák annak pedagógiai jelentőségét, hogy felismerjük és serkentsük a gyerekek önmaguk alkotta stratégiáinak rugalmasságát, ami fontos pillére az alsó tagozatos matematikatanítás innovatív megközelítésének. Készítettek és kipróbáltak olyan tananyagokat és fejlesztő programokat, amelyekkel ezt a fajta rugalmasságot kívánták növelni (lásd például Brownell, 1945; Freudenthal, 1991; Thompson, 1999; Wittmann és Müller, 1990-92). A múlt század végének számos reform-tantervében megjelenik a stratégia-rugalmasság kifejlesztésének pedagógiai értékéről alkotott meggyőződés

Utilizar las matemáticas para resolver problemas reales

En este trabajo de revisión se analizan las razones por las que los alumnos no son capaces de resolver problemas realistas utilizando conocimientos no matemáticos. Para ello se describen, en primer lugar, las investigaciones internacionales que han documentado estas dificultades en la resolución de problemas realistas. En segundo lugar, se describe cómo los libros de texto y la cultura del aula favorecen que los niños vayan aprendiendo de manera progresiva a resolver los problemas de matemáticas utilizando únicamente sus conocimientos matemáticos, y cómo este aprendizaje va abriendo una brecha entre las matemáticas escolares y el mundo real. En tercer lugar, se describe un estudio instruccional que ha demostrado que lograr que los alumnos resuelvan problemas realistas sin ceñirse exclusivamente a sus conocimientos matemáticos es un objetivo factible y deseable. Finalmente, se exponen una serie de consideraciones acerca del esfuerzo que supone este tipo de enseñanza de la matemática y la resolución de problemas y de los límites que marcan esta perspectiva del proceso de enseñanza y aprendizaje de las matemáticas.This paper analyses the reasons underlying children s difficulties when solving realistic wor(l)d problems in the mathematics class. In order to do so, first, international studies on these difficulties are reported. Second, we describe how characteristics of the current textbooks and classroom interaction led children to learn that mathematical problems must be solved by using only mathematical information and procedures, and not by attending to relevant real-world aspects of these problems, and how this learning process moves away the mathematics of the school from the real life. Third, an intervention study that showed how the classroom practice and culture can be successfully modified is described. Finally, some comments about the limits of these ideas are made

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

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

Why Humans Fail in Solving the Monty Hall Dilemma: A Systematic Review

The Monty Hall dilemma (MHD) is a difficult brain teaser. We present a systematic review of literature published between January 2000 and February 2018 addressing why humans systematically fail to react optimally to the MHD or fail to understand it. Based on a sequential analysis of the phases in the MHD, we first review causes in each of these phases that may prohibit humans to react optimally and to fully understand the problem. Next, we address the question whether humans’ performance, in terms of choice behaviour and (probability) understanding, can be improved. Finally, we discuss individual differences related to people’s suboptimal performance. This review provides novel insights by means of its holistic approach of the MHD: At each phase, there are reasons to expect that people respond suboptimally. Given that the occurrence of only one cause is sufficient, it is not surprising that suboptimal responses are so widespread and people rarely understand the MHD