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13 pages, 1372 KiB

Open AccessArticle

Analysis of Teacher-Student Interaction in the Joint Solving of Non-Routine Problems in Primary Education Classrooms

by Beatriz Sánchez-Barbero, José María Chamoso, Santiago Vicente and Javier Rosales

Sustainability 2020, 12(24), 10428; https://doi.org/10.3390/su122410428 - 13 Dec 2020

Cited by 10 | Viewed by 3174

The analysis of teacher–student interaction when jointly solving routine problems in the primary education mathematics classroom has revealed that there is scarce reasoning and little participation on students’ part. To analyze whether this fact is due to the routine nature of the problems, [...] Read more.

The analysis of teacher–student interaction when jointly solving routine problems in the primary education mathematics classroom has revealed that there is scarce reasoning and little participation on students’ part. To analyze whether this fact is due to the routine nature of the problems, a sample of teachers who solved, together with their students, a routine problem involving three questions with different cognitive difficulty levels (task 1) was analyzed, describing on which part of the problem-solving process (selection of information or reasoning) they focused their interaction. Results showed that they barely focused the interaction on reasoning, and participation of students was scarce, regardless of the cognitive difficulty of the question to be answered. To check whether these results could be due to the routine nature of the problem, a nonroutine problem (task 2) was solved by the same sample of teachers and students. The results revealed an increase in both reasoning and participation of students in processes that required complex reasoning. This being so, the main conclusion of the present study is that including nonroutine problem solving in the primary education classroom as a challenging task is a reasonable way to increase students’ ability to use their own reasoning to solve problems, and to promote greater teacher–student collaboration. These two aspects are relevant for students to become creative, critical, and reflective citizens. Full article

(This article belongs to the Special Issue Math Education and Problem Solving)

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20 pages, 7472 KiB

Open AccessArticle

Strategies of Pre-Service Early Childhood Teachers for Solving Multi-Digit Division Problems

by Zaira Ortiz-Laso and José-Manuel Diego-Mantecón

Sustainability 2020, 12(23), 10217; https://doi.org/10.3390/su122310217 - 7 Dec 2020

Cited by 2 | Viewed by 3069

Abstract

Unlike previous research, this study analyzes the strategies of pre-service early childhood teachers when solving multi-digit division problems and the errors they make. The sample included 104 subjects from a university in Spain. The data analysis was framed under a mixed-method approach, integrating [...] Read more.

Unlike previous research, this study analyzes the strategies of pre-service early childhood teachers when solving multi-digit division problems and the errors they make. The sample included 104 subjects from a university in Spain. The data analysis was framed under a mixed-method approach, integrating both quantitative and qualitative analyses. The results revealed that the traditional division algorithm was widely used in problems involving integers, but not so frequently applied to problems with decimal numbers. Often, number-based and algebraic strategies were employed as an alternative to the traditional algorithm, as the pre-service teachers did not remember how to compute it. In general, number-based strategies reached more correct solutions than the traditional algorithm, while the algebraic strategies did not usually reach any solution. Incorrect identifications of the mathematical model were normally related to an exchange of the dividend and divisor roles. Most pre-service teachers not only failed to compute the division, but also to interpret the obtained solution in the problem context. The study concludes that, during their schooling, students accessing the Degree in Early Childhood education have not acquired the necessary knowledge and skills to solve multi-digit division problems, and thus the entrance requirements at the university must be rethought. Full article

(This article belongs to the Special Issue Math Education and Problem Solving)

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22 pages, 1550 KiB

Open AccessArticle

Relation between Pupils’ Mathematical Self-Efficacy and Mathematical Problem Solving in the Context of the Teachers’ Preferred Pedagogies

by Vlastimil Chytrý, Janka Medová, Jaroslav Říčan and Jiří Škoda

Sustainability 2020, 12(23), 10215; https://doi.org/10.3390/su122310215 - 7 Dec 2020

Cited by 7 | Viewed by 4061

Abstract

In research focused on self-efficacy it is usually teacher-related phenomena that are studied, while the main aspects related to pupils are rather neglected, although self-efficacy itself is perceived as a belief in one’s own abilities. Evidently, this strongly influences the behavior of individuals [...] Read more.

In research focused on self-efficacy it is usually teacher-related phenomena that are studied, while the main aspects related to pupils are rather neglected, although self-efficacy itself is perceived as a belief in one’s own abilities. Evidently, this strongly influences the behavior of individuals in terms of the goal and success in mathematical problem-solving. Considering that alternative teaching methods are based on the principle of belief in one’s own ability (mainly in the case of group work), higher self-efficacy can be expected in the pupils of teachers who use predominantly the well-working pupil-centered pedagogies. A total of 1133 pupils in grade 5 from 36 schools in the Czech Republic were involved in the testing of their ability to solve mathematical problems and their mathematical self-efficacy as well. Participants were divided according to the above criteria as follows: (i) 73 from Montessori primary schools, (ii) 332 pupils educated in mathematics according to the Hejný method, (iii) 510 pupils from an ordinary primary school, and (iv) 218 pupils completing the Dalton teaching plan. In the field of mathematical problem-solving the pupils from the Montessori primary schools clearly outperformed pupils from the Dalton Plan schools (p = 0.027) as well as pupils attending ordinary primary schools (p = 0.009), whereas the difference between the Montessori schools and Hejný classes was not significant (p = 0.764). There is no statistically significant difference in the level of self-efficacy of pupils with respect to the preferred strategies for managing learning activities (p = 0.781). On the other hand, correlation between mathematical problem-solving and self-efficacy was confirmed in all the examined types of schools. However, the correlation coefficient was lower in the case of the pupils from the classes applying the Hejný method in comparison with the pupils attending the Montessori schools (p = 0.073), Dalton Plan schools (p = 0.043), and ordinary primary schools (p = 0.002). Even though the results in mathematical problem-solving are not consistent across the studies, the presented results confirm better performance of pupils in some constructivist settings, particularly in the case of individual constructivism in the Montessori primary schools. The factors influencing lower correlation of self-efficacy and performance in mathematical problem-solving ought to be subject to further investigation. Full article

(This article belongs to the Special Issue Math Education and Problem Solving)

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28 pages, 3998 KiB

Open AccessArticle

Examples of Problem-Solving Strategies in Mathematics Education Supporting the Sustainability of 21st-Century Skills

by Zsuzsanna Katalin Szabo, Péter Körtesi, Jan Guncaga, Dalma Szabo and Ramona Neag

Sustainability 2020, 12(23), 10113; https://doi.org/10.3390/su122310113 - 3 Dec 2020

Cited by 59 | Viewed by 24833

Abstract

The overall aim of education is to train proactive, motivated, and independent citizens to face and overcome continuous challenges. Critical thinking—finding solutions to problems—is of primary importance in the 21st century to handle challenging situations and deal with obstacles in careers. A critical [...] Read more.

The overall aim of education is to train proactive, motivated, and independent citizens to face and overcome continuous challenges. Critical thinking—finding solutions to problems—is of primary importance in the 21st century to handle challenging situations and deal with obstacles in careers. A critical literature review approach was used to assess, critique, synthesizes, and expand the theoretical foundation of the topic. Teaching mathematical problem-solving is an efficient way to develop 21st-century skills and to give cross-curricular experiences with real-world meaning to learners. Concrete examples were presented to prove that Pólya’s heuristic could be used in a broader context to help learners acquire the modern skills needed to succeed in their careers. By including in the learning process and practicing specific methods for solving mathematical problems, students could learn a way of thinking to approach and solve problems successfully in a broader context in life. The paper’s outcome provides teachers and educators with methods, learning models, and strategies for developing 21st-century skills in students at all levels during classroom activities. Full article

(This article belongs to the Special Issue Math Education and Problem Solving)

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17 pages, 2606 KiB

Open AccessArticle

Creating Realistic Mathematics Tasks Involving Authenticity, Cognitive Domains, and Openness Characteristics: A Study with Pre-Service Teachers

by Sara Paredes, María José Cáceres, José-Manuel Diego-Mantecón, Teresa F. Blanco and José María Chamoso

Sustainability 2020, 12(22), 9656; https://doi.org/10.3390/su12229656 - 19 Nov 2020

Cited by 13 | Viewed by 3511

Abstract

Creating mathematics tasks provide opportunities for students to develop their thinking, reasoning, communication, and creativity. This paper presents a study on teaching pre-service teachers to create realistic mathematics tasks in real contexts and amending them through an iterative process of analysis and refinement. [...] Read more.

Creating mathematics tasks provide opportunities for students to develop their thinking, reasoning, communication, and creativity. This paper presents a study on teaching pre-service teachers to create realistic mathematics tasks in real contexts and amending them through an iterative process of analysis and refinement. The study was undertaken with pre-service teachers from two university training courses in Spain, undergraduate students from a primary teacher training course, and graduate students from an educational Master’s course. The students worked in groups to collaborate in the creation of the requested tasks and improvement of them based on critical thinking and creativity. The tasks were not only evaluated concerning their level of realism, but also regarding their level of authenticity, the cognitive domains involved, and their openness characteristic. These are the key characteristics related to environmental and sustainability aspects. The outcomes confirmed that the creation of realistic mathematics tasks was a challenge for future primary teachers; however, they were able to create tasks with high levels of cognitive domain, authenticity, and openness. This evidences, on the one hand, the difficulty that future teachers have in understanding the realism of a mathematics task, and, on the other, the possibilities offered by the task’s creation and the revision activity, which has educational implications and opens paths for future research. Full article

(This article belongs to the Special Issue Math Education and Problem Solving)

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21 pages, 1355 KiB

Open AccessArticle

Subtraction: More than an Algorithm?

by M. Mercedes Rodríguez-Sánchez, Ana B. Sánchez-García and Ricardo López-Fernández

Sustainability 2020, 12(21), 9148; https://doi.org/10.3390/su12219148 - 3 Nov 2020

Cited by 1 | Viewed by 6169

Abstract

One of the aims of compulsory education is for students to adequately handle basic maths, owing to its importance in their future professional and personal lives. However, mechanical knowledge of an algorithm may not be sufficient to train future citizens with critical and [...] Read more.

One of the aims of compulsory education is for students to adequately handle basic maths, owing to its importance in their future professional and personal lives. However, mechanical knowledge of an algorithm may not be sufficient to train future citizens with critical and creative thinking if it is not accompanied by a comprehensive understanding of the concept. In this regard, existing research shows that a high percentage of students in primary education commit errors when they attempt subtraction. However, little is known about whether adults perform the same calculations correctly. In this context, 535 university students completed a questionnaire composed of 20 subtractions. The results showed that only one quarter of respondents performed the subtractions correctly. Analysis of error type showed that the most frequent mistakes corresponded to the systematic errors made by primary-level students. This may indicate that the types of errors committed during early learning persist over time, implying that subtraction may not have been adequately taught. New educational approaches and initiatives are required to encourage the teaching and learning of subtraction in a more reasoned and critical manner during early learning. Full article

(This article belongs to the Special Issue Math Education and Problem Solving)

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18 pages, 2674 KiB

Open AccessArticle

Characterisers of Teaching in a Mathematics Problem Posing Lesson in Preschool Education

by Juan Pedro Martín-Díaz, Miguel Montes, Myriam Codes and José Carrillo

Sustainability 2020, 12(15), 6148; https://doi.org/10.3390/su12156148 - 30 Jul 2020

Cited by 8 | Viewed by 2409

Abstract

Problem posing is rarely seen in Spanish classrooms, least of all in preschool education. The lack of research, coupled with a lack of guidance, has resulted in teachers simply not considering the approach as something that could enrich their work. This study examines [...] Read more.

Problem posing is rarely seen in Spanish classrooms, least of all in preschool education. The lack of research, coupled with a lack of guidance, has resulted in teachers simply not considering the approach as something that could enrich their work. This study examines a problem-posing lesson given by a preschool teacher to a class of four-year-olds. It focuses in particular on the organisational principles and the factors to be taken into consideration in applying the strategy. Through the application of a bottom-up analysis, seven characterisers are identified, four associated with task design and planning: Characteristics of the resources, type of problem, expression of problem formulation, and type of problem formulation; and three associated with lesson management: Stages in task execution, degree of liberty, and role of the pupils’ drawings of the problem. Full article

(This article belongs to the Special Issue Math Education and Problem Solving)

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12 pages, 10603 KiB

Open AccessArticle

Generating Worthwhile Mathematical Tasks in Order to Sustain and Develop Mathematical Thinking

by John Mason

Sustainability 2020, 12(14), 5727; https://doi.org/10.3390/su12145727 - 16 Jul 2020

Cited by 5 | Viewed by 3218

Abstract

Making use of a phenomenological stance which first and foremost values the lived experience of learners, six tasks are used to illustrate what it might mean for a mathematical task to be deemed worthy of being offered to learners. These take the form [...] Read more.

Making use of a phenomenological stance which first and foremost values the lived experience of learners, six tasks are used to illustrate what it might mean for a mathematical task to be deemed worthy of being offered to learners. These take the form of encounters with, and opportunities to develop, pervasive mathematical themes, use of mathematical powers and experience of mathematical concepts and topics. Comments about how worthwhile mathematical tasks can evolve centre around developing the propensity, the habit of mind to extend, vary and generalise for oneself. Mathematical thinking is sustained by developing this disposition. Full article

(This article belongs to the Special Issue Math Education and Problem Solving)

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