The relationship between mathematical problem-solving skills and self-regulated learning through homework behaviours,motivation, and metacognition

Studies highlight that using appropriate strategies during problem solving is important to improve problem-solving skills and draw attention to the fact that using these skills is an important part of students’ self-regulated learning ability. Studies on this matter view the self-regulated learning ability as key to improving problem-solving skills. The aim of this study is to investigate the relationship between mathematical problem-solving skills and the three dimensions of self-regulated learning (motivation, metacognition, and behaviour), and whether this relationship is of a predictive nature. The sample of this study consists of 323 students from two public secondary schools in Istanbul. In this study, the mathematics homework behaviour scale was administered to measure students’ homework behaviours. For metacognition measurements, the mathematics metacognition skills test for students was administered to measure offline mathematical metacognitive skills, and the metacognitive experience scale was used to measure the online mathematical metacognitive experience. The internal and external motivational scales used in the Programme for International Student Assessment (PISA) test were administered to measure motivation. A hierarchic regression analysis was conducted to determine the relationship between the dependent and independent variables in the study. Based on the findings, a model was formed in which 24% of the total variance in students’ mathematical problem-solving skills is explained by the three sub-dimensions of the self-regulated learning model: internal motivation (13%), willingness to do homework (7%), and post-problem retrospective metacognitive experience (4%).     

Is there a causal relation between mathematical creativity and mathematical problem-solving performance?

The relationship between mathematical creativity (MC) and mathematical problem-solving performance (MP) has often been studied but the causal relation between these two constructs has yet to be clearly reported. The main purpose of this study was to define the causal relationship between MC and MP. Data from a representative sample of 480 eighth-grade students were analysed using a cross-lagged panel correlation (CLPC) design. CLPC attempts to rule out plausible alternative explanation of a causal effect. The result suggests that significant predominant causal relationship was found between MC and MP. It indicates that MP was found to be a cause of MC than the converse.     

A naturalistic study of executive function and mathematical problem-solving

Our goal in this research was to understand the specific challenges middle-school students face when engaging in mathematical problem-solving by using executive function (i.e., shifting, updating, and inhibiting) of working memory as a functional construct for the analysis. Using modified talk-aloud protocols, real-time naturalistic analysis of eighth-grade students’ mathematical problem-solving were conducted. A fine-grained coding of the students’ talking-aloud during problem-solving in mathematics involved isolating the challenges students faced in each one of the four problem-solving phases, and then making a functional link to one of the executive functions of shifting, updating, and inhibiting. In total, 344 episodes were analyzed. Our results show that updating proved to be most challenging during the understanding the problem phase, inhibiting during the carrying out the plan phase, and shifting during the looking back and evaluation phase. Furthermore, students are more likely to make progress with the problem-solving if they are able to engage in a conscious appraisal of the problem at the onset of the problem-solving. Assisting students in establishing what the problem requires through the cognitive clues presented in the problem may necessitate explicit instructional on behalf of the teacher.     

A metacognitive-based instruction for Primary Four students to approach non-routine mathematical word problems

Many Primary Four students (fourth graders) in Singapore have difficulties initiating or persevering in the problem-solving process even though the curriculum has focused on problem solving since 1992. This study served to examine the role of metacognition in self-regulated problem solving. The study, a quasi-experimental pretest–posttest design involving a convenience sample of 63 students from two intact mixed-ability Primary Four classes, examined the impact of using a metacognitive scheme that focuses on the understanding and planning stages of Pòlya’s four-stage approach on students’ mathematical problem-solving behavior, performance and attitudes. The findings revealed that the metacognitive-based scheme had a positive impact on students’ understanding of the problem posed, solution planning, confidence in and personal control of problem-solving behavior and emotions. It had also helped them to initiate and persevere in the problem-solving process to achieve a higher level of problem-solving success. Limitations and instructional implications are discussed.     

Opportunity to learn problem solving in Dutch primary school mathematics textbooks

In the Netherlands, mathematics textbooks are a decisive influence on the enacted curriculum. About a decade ago, Dutch primary school mathematics textbooks provided hardly any opportunities to learn problem solving. In this study we investigated whether this provision has changed. In order to do so, we carried out a textbook analysis in which we established to what degree current textbooks provide non-routine problem-solving tasks for which students do not immediately have a particular solution strategy at their disposal. We also analyzed to what degree textbooks provide ‘gray-area’ tasks, which are not really non-routine problems, but are also not straightforwardly solvable. In addition, we inventoried other ways in which present textbooks facilitate the opportunity to learn problem solving. Finally, we researched how inclusive these textbooks are with respect to offering opportunities to learn problem solving for students with varying mathematical abilities. The results of our study show that the opportunities that the currently most widely used Dutch textbooks offer to learn problem solving are very limited, and these opportunities are mainly offered in materials meant for more able students. In this regard, Dutch mainstream textbooks have not changed compared to the situation a decade ago. A textbook that is the Dutch edition of a Singapore mathematics textbook stands out in offering the highest number of problem-solving tasks, and in offering these in the materials meant for all students. However, in the ways this textbook facilitates the opportunity to learn problem solving, sometimes a tension occurs concerning the creative character of genuine problem solving.     

Mathematical literacy in undergraduates: role of gender,emotional intelligence and emotional self-efficacy

This empirical study explores the roles that Emotional Intelligence (EI) and Emotional Self-Efficacy (ESE) play in undergraduates’ mathematical literacy, and the influence of EI and ESE on students’ attitudes towards and beliefs about mathematics. A convenience sample of 93 female and 82 male first-year undergraduates completed a test of mathematical literacy, followed by an online survey designed to measure the students’ EI, ESE and factors associated with mathematical literacy. Analysis of the data revealed significant gender differences. Males attained a higher mean test score than females and out-performed the females on most of the individual questions and the associated mathematical tasks. Overall, males expressed greater confidence in their mathematical skills, although both males’ and females’ confidence outweighed their actual mathematical proficiency. Correlation analyses revealed that males and females attaining higher mathematical literacy test scores were more confident and persistent, exhibited lower levels of mathematics anxiety and possessed higher mathematics qualifications. Correlation analyses also revealed that in male students, aspects of ESE were associated with beliefs concerning the learning of mathematics (i.e. that intelligence is malleable and that persistence can facilitate success), but not with confidence or actual performance. Both EI and ESE play a greater role with regard to test performance and attitudes/beliefs regarding mathematics amongst female undergraduates; higher EI and ESE scores were associated with higher test scores, while females exhibiting higher levels of ESE were also more confident and less anxious about mathematics, believed intelligence to be malleable, were more persistent and were learning goal oriented. Moderated regression analyses confirmed mathematics anxiety as a negative predictor of test performance in males and females, but also revealed that in females EI and ESE moderate the effects of anxiety on test performance, with the relationship between anxiety and test performance linked more to emotional management (EI) than to ESE.     

Problem modification as a tool for detecting cognitive flexibility in school children

This paper presents the results of an experiment in which fourth to sixth graders with above-average mathematical abilities modified a given problem. The experiment found evidence of links between problem posing and cognitive flexibility. Emerging from organizational theory, cognitive flexibility is conceptualized through three primary constructs: cognitive variety, cognitive novelty, and changes in cognitive framing. Among these components, changes in cognitive framing could be effectively detected in problem-posing situations, giving a relevant indication of students’ creative potential. The students’ capacity to generate coherent and consistent problems in the context of problem modification may indicate the existence of a strategy of functional type for generalizations, which seems to be specific to mathematical creativity.     

Connecting mathematical creativity to mathematical ability

This study aims to investigate whether there is a relationship between mathematical ability and mathematical creativity, and to examine the structure of this relationship. Furthermore, in order to validate the relationship between the two constructs, we will trace groups of students that differ across mathematical ability and investigate the relationships amongst these students’ performance on a mathematical ability test and the components of mathematical creativity. Data were collected by administering two tests, a mathematical ability and a mathematical creativity test, to 359 elementary school students. Mathematical ability was considered as a multidimensional construct, including quantitative ability (number sense and pre-algebraic reasoning), causal ability (examination of cause–effect relations), spatial ability (paper folding, perspective and spatial rotation abilities), qualitative ability (processing of similarity and difference relations) and inductive/deductive ability. Mathematical creativity was defined as a domain-specific characteristic, enabling individuals to be characterized by fluency, flexibility and originality in the domain of mathematics. The data analysis revealed that there is a positive correlation between mathematical creativity and mathematical ability. Moreover, confirmatory factor analysis suggested that mathematical creativity is a subcomponent of mathematical ability. Further, latent class analysis showed that three different categories of students can be identified varying in mathematical ability. These groups of students varying in mathematical ability also reflected three categories of students varying in mathematical creativity.     

The potential of recreational mathematics to support the development of mathematical learning

ABSTRACT

A literature review establishes a working definition of recreational mathematics: a type of play which is enjoyable and requires mathematical thinking or skills to engage with. Typically, it is accessible to a wide range of people and can be effectively used to motivate engagement with and develop understanding of mathematical ideas or concepts. Recreational mathematics can be used in education for engagement and to develop mathematical skills, to maintain interest during procedural practice and to challenge and stretch students. It can also make cross-curricular links, including to history of mathematics. In undergraduate study, it can be used for engagement within standard curricula and for extra-curricular interest. Beyond this, there are opportunities to develop important graduate-level skills in problem-solving and communication. The development of a module ‘Game Theory and Recreational Mathematics’ is discussed. This provides an opportunity for fun and play, while developing graduate skills. It teaches some combinatorics, graph theory, game theory and algorithms/complexity, as well as scaffolding a Pólya-style problem-solving process. Assessment of problem-solving as a process via examination is outlined. Student feedback gives some indication that students appreciate the aims of the module, benefit from the explicit focus on problem-solving and understand the active nature of the learning.     

Student Difficulties with Accessing and Using Mathematical Knowledge

This article examines the issue of why students fail to activate and use mathematical knowledge during problem solving when it is known that they possess the required knowledge. This issue is explored by analyzing problem-solving attempts of a high-achieving student and a low-achieving student in the domain of plane geometry. On the basis of these data and other literature, three major sources of mathematical knowledge-access difficulties are identified that might be considered by classroom teachers, including a student's (1) dispositional state, (2) management of the problem-solving process, and (3) state of organization of his or her mathematical knowledge. It is argued that teaching practices that place emphasis on careful management of problem-solving activity could help students activate and extend the use of mathematical knowledge acquired in lesson activities.     

A comparison of mathematical features of Turkish and American textbook definitions regarding special quadrilaterals

In this study, I analyzed and compared definitions of special quadrilaterals presented in the US and Turkish secondary school mathematics textbooks. To this end, seven textbooks from each country were examined based on the following features of mathematical definitions: form of presentation and mathematical correctness, minimality, the nature of definitions, and the reference sets and geometric properties used in definitions. The results showed that Turkish textbooks include both procedural and structural definitions of quadrilaterals, while procedural definitions are non-existent in the US mathematics textbooks. Meanwhile, all structural definitions and all procedural definitions excluding one are mathematically correct. Besides, in both country textbooks, there exists more number of minimal and inclusive definitions compared to the non-minimal and exclusive ones. Finally, textbooks from both countries mainly use quadrilaterals as reference sets and sides as geometric properties in definitions of special quadrilaterals. The results bring to light the varied opportunities provided to the students by the US and Turkish textbooks to experience and learn definitions of special quadrilaterals.     

Mathematical reasoning in teachers’ presentations

This paper presents a study of the opportunities presented to students that allow them to learn different types of mathematical reasoning during teachers’ ordinary task solving presentations. The characteristics of algorithmic and creative reasoning that are seen in the presentations are analyzed. We find that most task solutions are based on available algorithms, often without arguments that justify the reasoning, which may lead to rote learning. The students are given some opportunities to see aspects of creative reasoning, such as reflection and arguments that are anchored in the mathematical properties of the task components, but in relatively modest ways.     

Some environmental and attitudinal characteristics as predictors of mathematical creativity

There are many things which can be made more useful and interesting through the application of creativity. Self-concept in mathematics and some school environmental factors such as resource adequacy, teachers’ support to the students, teachers’ classroom control, creative stimulation by the teachers, etc. were selected in the study. The sample of the study comprised 770 seventh grade students. Pearson correlation, multiple correlation, regression equation and multiple discriminant function analyses of variance were used to analyse the data. The result of the study showed that the relationship between mathematical creativity and each attitudinal and environmental characteristic was found to be positive and significant. Index of forecasting efficiency reveals that mathematical creativity may be best predicted by self-concept in mathematics. Environmental factors, resource adequacy and creative stimulation by the teachers’ are found to be the most important factors for predicting mathematical creativity, while social–intellectual involvement among students and educational administration of the schools are to be suppressive factors. The multiple correlation between mathematical creativity and attitudinal and school environmental characteristic suggests that the combined contribution of these variables plays a significant role in the development of mathematical creativity. Mahalanobis analysis indicates that self-concept in mathematics and total school environment were found to be contributing significantly to the development of mathematical creativity.     

Enhancing student engagement to positively impact mathematics anxiety,confidence and achievement for interdisciplinary science subjects

Contemporary science educators must equip their students with the knowledge and practical know-how to connect multiple disciplines like mathematics, computing and the natural sciences to gain a richer and deeper understanding of a scientific problem. However, many biology and earth science students are prejudiced against mathematics due to negative emotions like high mathematical anxiety and low mathematical confidence. Here, we present a theoretical framework that investigates linkages between student engagement, mathematical anxiety, mathematical confidence, student achievement and subject mastery. We implement this framework in a large, first-year interdisciplinary science subject and monitor its impact over several years from 2010 to 2015. The implementation of the framework coincided with an easing of anxiety and enhanced confidence, as well as higher student satisfaction, retention and achievement. The framework offers interdisciplinary science educators greater flexibility and confidence in their approach to designing and delivering subjects that rely on mathematical concepts and practices.     

Fostering creativity through instruction rich in mathematical problem solving and problem posing

Although creativity is often viewed as being associated with the notions of “genius” or exceptional ability, it can be productive for mathematics educators to view creativity instead as an orientation or disposition toward mathematical activity that can be fostered broadly in the general school population. In this article, it is argued that inquiry-oriented mathematics instruction which includes problem-solving and problem-posing tasks and activities can assist students to develop more creative approaches to mathematics. Through the use of such tasks and activities, teachers can increase their students’ capacity with respect to the core dimensions of creativity, namely, fluency, flexibility, and novelty. Because the instructional techniques discussed in this article have been used successfully with students all over the world, there is little reason to believe that creativity-enriched mathematics instruction cannot be used with a broad range of students in order to increase their representational and strategic fluency and flexibility, and their appreciation for novel problems, solution methods, or solutions.     

Exploring the relationship between metacognitive and collaborative talk during group mathematical problem-solving – what do we mean by collaborative metacognition?

The purpose of this study was to enhance our understanding of the relationship between collaborative talk and metacognitive talk during group mathematical problem-solving. Research suggests that collaborative talk may mediate the use of metacognitive talk, which in turn is associated with improved learning outcomes. However, our understanding of the role of group work on the individual use of metacognition during problem-solving has been limited because research has focused on either the individual or the group as a collective. Here, primary students (aged nine to 10) were video-recorded in a naturalistic classroom setting during group mathematical problem-solving sessions. Student talk was coded for metacognitive, cognitive and social content, and also for collaborative content. Compared with cognitive talk, we found that metacognitive talk was more likely to meet the criteria to be considered collaborative, with a higher probability of being both preceded by and followed by collaborative talk. Our results suggest that collaborative metacognition arises from combined individual and group processes.     

Conceptualizing Perseverance in Problem Solving as Collective Enterprise

Students are expected to learn mathematics such that when they encounter challenging problems they will persist. Creating opportunities for students to persist in problem solving is therefore argued as essential to effective teaching and to children developing positive dispositions in mathematical learning. This analysis takes a novel approach to perseverance by conceptualizing it as collective enterprise among learners in lieu of its more conventional treatment as an individual capacity. Drawing on video of elementary school children in two US classrooms (n = 52), this paper offers: (1) empirical examples that define perseverance as collective enterprise; (2) indicators of perseverance for teachers (and researchers) to support (and study) its emergence; and (3) evidence of how the task, peer dynamics, and student-teacher interactions afford or constrain its occurrence. The significance of perseverance as collective enterprise and as an object of design in developing effective learning communities, is discussed.     

Mathematical poetic structures: The sound shape of collaboration

This paper outlines a new method of mathematical discourse analysis focused on identifying poetic structures in students’ mathematical conversations. Following the linguistic anthropology tradition inspired by Roman Jakobson, poetic structures refer to any conversational repetition of sounds, words or syntax; this repetition draws attention to the form of the message. In mathematical conversations, poetic structures can express patterns, rhythms, similarities or dissimilarities associated with a task. Methodological dilemmas associated with identifying and representing poetic structures and pragmatic responses are highlighted. An analysis of a nine minute algebraic problem-solving conversation revealed eight types of mathematical poetic structures that collectively assisted all of the students’ vital mathematical insights. The paper aims to demonstrate that poetic analysis of mathematical conversations can bridge the illusory distinction between mathematical discourse and mathematical reasoning.