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%).     

Spatial visualizers, object visualizers and verbalizers: their mathematical creative abilities

This paper investigates the relationship between the creative process in mathematical tasks and spatial, object and verbal cognitive styles. A group of 96 prospective primary school teachers completed the Object-Spatial Imagery and Verbal Questionnaire and took a mathematical creativity test. The results of a multiple regression analysis demonstrated that whereas visual cognitive styles (spatial and object imagery) were statistically significant predictors of participants’ creative abilities in mathematics, verbal cognitive style did not predict these abilities. Further analysis of the data indicated that spatial imagery cognitive style was related to mathematical fluency, flexibility and originality. On the other hand, object imagery cognitive style was negatively related to mathematical originality and verbal cognitive style was negatively related to mathematical flexibility. The study also revealed that individuals with a tendency towards different cognitive styles employed different strategies in the creative mathematical tasks.     

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.     

Structuring mathematical knowledge and skills by means of knowledge graphs

The aim of the study reported on in this paper was to develop, test and improve a cognitive tool which could help students structure their mathematical knowledge and skills. Mathematics teaching as an auxiliary subject in the context of secondary or tertiary education courses in other disciplines pays too little attention to the structure of the mathematical concepts presented. For the students, therefore, the network of relationships between these concepts does not become a part of their mathematical knowledge and skills, and is consequently not fully available for purposes of reasoning, proving, mathematicizing and solving problems. Knowledge graphs (KGs) can be used by students as a tool to visualize this structure of the concepts and the relations between them. The learning activity of structuring one's mathematical knowledge and skills can be supported by a model, the Mathematical Knowledge Graph Model (MKGM), which serves as a pre-structured heuristic framework. The elements of this model include a central concept, special cases of this concept, operations or actions on the concept, areas of application and properties of the concepts and operations. The present paper reports on a trial among five students of the Open University of the Netherlands (OUNL), who constructed a KG in accordance with the MKGM model. The paper focuses on the graphs produced by the students, their appreciation of the structuring activity and the relation between their graphs and test results.     

How teaching to foster mathematical creativity may impact student self-efficacy for proving

Mathematical creativity has been emphasized as an essential part of mathematics, yet little research has been done to study the effects of fostering creativity in the tertiary mathematics classroom. In this paper, we explore how fostering mathematical creativity may impact student self-efficacy for proving. For this, we developed new methods to study evidence of instructor use of Sriraman’s (2005) five principles for fostering mathematical creativity and changes in student self-efficacy via Bandura's (1997) four sources of self-efficacy. This revealed associations between four of the five principles and changes in student self-efficacy for proving, along with two instances where the combined use of principles may have provided students greater opportunities for building self-efficacy for proving. The implications of these results for teaching and future research are discussed.     

Development of a mathematical ability test: a validity and reliability study

The identification of talented students accurately at an early age and the adaptation of the education provided to the students depending on their abilities are of great importance for the future of the countries. In this regard, this study aims to develop a mathematical ability test for the identification of the mathematical abilities of students and the determination of the relationships between the structure of abilities and these structures. Furthermore, this study adopts test development processes. A structure consisting of the factors of quantitative ability, causal ability, inductive/deductive reasoning ability, qualitative ability and spatial ability has been obtained following this study. The fit indices of the finalized version of the mathematical ability test of 24 items indicate the suitability of the test.     

Mathematical creativity in generally gifted and mathematically excelling adolescents: what makes the difference?

Due to uncertainty regarding the relationship between mathematical creativity, mathematical expertise and general giftedness, we have conducted a large-scale study that explores the relationship between mathematical creativity and mathematical ability. We distinguish between relative and absolute creativity in order to address personal creativity as a characteristic that can be developed in schoolchildren. This paper presents part of a study that focuses on the power of multiple solution tasks (MSTs) as a tool for the evaluation of relative creativity. We discuss relationships between mathematical creativity, mathematical ability and general giftedness as reflected in the present empirical study of senior high school students in Israel which implemented the MST tool. The study demonstrates that between-group differences are task dependent and are a function of mathematical insight as it is integrated in the mathematical task. Thus, we conclude that different types of MSTs can be used for different research purposes, which we discuss at the end of this paper.     

How do students’ behaviors relate to the growth of their mathematical ideas?

The purpose of this study is to analyze the relationship between student behaviors and the growth of mathematical ideas (using the Pirie-Kieren model). This analysis was accomplished through a series of case studies, involving middle school students of varying ability levels, who were investigating a combinatorics problem in after-school problem-solving sessions. The results suggest that certain types of student behaviors appear to be associated with the growth of ideas and emerge in specific patterns. More specifically, as understanding grows, there is a general shift from behaviors such as students questioning each other, explaining and using their own and others’ ideas toward behaviors involving the setting up of hypothetical situations, linking of representations and connecting of contexts. Recognizing that certain types of student behaviors tend to emerge in specific layers of the Pirie-Kieren model can be important in helping us to understand the development of mathematical ideas in children.     

Recognising mathematical creativity in schoolchildren

Examples of tasks designed to recognise creative thinking within mathematics, used with 11–12-year-old pupuls, are described. The first construct empoyed in the design of these tasks is the ability to overcome fixation. Sometimes pupils demonstrate content-universe fixation, by restricting their thinking about a problem to an insufficient or inappropriate range of elements. Other times they show algorithmic fixation by continuing to adhere to a routine procedure or stereotype response even when this becomes inefficient or inappropriate. The second construct employed is that of divergent production, indicated by flexibility and originality in mathematical tasks to which a large number of appropriate responses are possible. Examples of three categories of such tasks are described: (1) problem-solving, (2) problem-posing, and (3) redefinition. Examples of pupils’ responses to various tasks are used to argue that they do indeed reveal thinking that can justifiably be described as creative. The relationship to conventional mathematics attainment is discussed-mathematics attainment is seen to limit but not to determine mathematical creativity.     

New challenges in the 2011 revised middle school curriculum of South Korea: mathematical process and mathematical attitude

The ‘future-oriented middle school mathematics curriculum focused on creativity and personality’ was revised in August of 2011 with the aim of nurturing students’ mathematical creativity and sound personalities. The curriculum emphasizes: contextual learning from which students can grasp mathematical concepts and make connections with their everyday lives; manipulation activities through which students may attain an intuitive idea of what they are learning and enhance their creativity; and reasoning to justify mathematical results based on their knowledge and experience. Since students will not be able to engage in the intended mathematical process with the study-load imposed by the current curriculum, the newly revised curriculum modifies or deletes some parts of the contents that have been traditionally taught mechanically. This paper provides a detailed overview of the main points of the revised curriculum.     

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.     

Mathematics and engineering in real life through mathematical competitions

We bring out an experience of organizing mathematical competitions that can be used as a medium to motivate the student and teacher minds in new directions of thinking. This can contribute to fostering research, innovation and provide a hands-on experience of mathematical concepts with the real world. Mathematical competitions can be used to build curiosity and give an understanding of mathematical applications in real life. Participation in the competition has been classified under four broad categories. Student can showcase their findings in various forms of expression like model, poster, soft presentation, animation, live performance, art and poetry. The basic focus of the competition is on using open source computation tools and modern technology, to emphasize the relationship of mathematical concepts with engineering applications in real life.     

Teachers’ mathematical activity in inquiry-oriented instruction

This work investigates the relationship between teachers’ mathematical activity and the mathematical activity of their students. By analyzing the classroom video data of mathematicians implementing an inquiry-oriented abstract algebra curriculum I was able to identify a variety of ways in which teachers engaged in mathematical activity in response to the mathematical activity of their students. Further, my analysis considered the interactions between teachers’ mathematical activity and the mathematical activity of their students. This analysis suggests that teachers’ mathematical activity can play a significant role in supporting students’ mathematical development, in that it has the potential to both support students’ mathematical activity and influence the mathematical discourse of the classroom community.     

Connecting multiplicative thinking and mathematical reasoning in the middle years

Mathematical reasoning and problem solving are recognised as essential 21st century skills. However, international assessments of mathematical literacy suggest these are areas of difficulty for many students. Evidenced-based learning trajectories that identify the key ideas and strategies needed to teach mathematics for understanding and support these important capacities over time are needed to support teachers and curriculum developers so that they do not have to rely solely on mathematics content knowledge. Given this goal and recent evidence to suggest a relationship between the development of multiplicative thinking and mathematical reasoning, this paper explores the processes involved in developing a single, integrated scale for multiplicative thinking and mathematical reasoning using data from a four-year design-based project to establish learning and assessment frameworks for algebraic, geometrical and statistical reasoning in the middle years of schooling.     

Comparisons of mathematical creativity,some personality and biographical factors of middle school dropouts and stayins

Middle school dropouts and stayins were compared on mathematical creativity, some personality and biographical factors. Verbal and non‐verbal mathematical creativity tests, a Hindi adaptation of the Thorndike dimensions of temperament test and a biographical inventory were used on 70 dropouts and 100 stayins male students, aged 11+ to 13+ years, randomly selected, from Sultanpur District, India. The results showed that: (1) mathematical creativity of dropouts was found to be lower than stayins; (2) dropouts were found to be sociable, accepting, reflective, lethargic and casual in nature whereas stayins were found to be solitary, critical, practical, premeditated, active and responsible in nature; and (3) the level of family income, professional background of the family, parents’ education, standard of living, interest‐patterns, attitude and level of aspiration of the stayins were found to be higher than of the dropouts.     

Cognitive styles,task presentation mode and mathematical performance

This paper reports the outcomes of an empirical study undertaken to investigate the relationship of prospective teachers’ cognitive styles and levels of performance in measurement and spatial tasks. A total of 116 prospective kindergarten school teachers were tested using the VICS and the extended CSA-WA tests (Peterson 2005) in order to place them along the Verbal/Imagery and the Wholistic/Analytic cognitive style continua. The same prospective teachers were also administered a mathematical test with 6 measurement and 6 spatial tasks. The results suggest that there were no significant differences between Verbalisers-Imagers and Wholistic-Analytic prospective teachers in their performance on the spatial pictorial and textual tasks, and on the measurement textual tasks. However, there were differences between Verbalisers-Imagers and Wholistic-Analytic prospective teachers in their performance on the measurement pictorial tasks. This difference was attributed to the performance of low achievers. High achievers performed in the same way independently of their cognitive styles.     

What Students Say About Their Mathematical Thinking When They Listen

Mathematical listening is an important aspect of mathematical communication. Yet there are relatively few examinations of this phenomenon. Further, existing studies of students' mathematical listening come from observational data, lacking the student perspective. This study examined student replies to an open‐response question regarding what happens to their thinking about mathematics when they listen to their peers' mathematical talk. Results suggest varying ways of listening that range from more passive to more active forms. While relationships were observed between ways of listening and perceived forms of engaging in mathematical discussion, no relationship was observed with the frequency with which students reported participating in discussions.     

Development and Analysis of a Mathematics Aptitude Test for Gifted Elementary School Students

This paper describes the development and preliminary analysis of a mathematical test targeted for high mathematical ability elementary school students, the Stanford Education Program for Gifted Youth (EPGY) Mathematical Aptitude Test (SEMAT). A version was administered to 248 students, 9–11 years old, in EPGY. The SEMAT was developed because no other satisfactory test was designed or normed for this population. Most standardized tests assess mathematics proficiency for the general population so that gifted students' scores cluster in the few top percentiles. The SEMAT discriminated among this extreme upper end. Item response theory determined proficiency estimates, which were then used as scores to predict various outcomes in EPGY. The SEMAT proved to be a strong predictor of acceleration in EPGY.