Group metacognition during mathematical problem solving

Although various studies have shown that groups are more productive than individuals in complex mathematical problem solving, not all groups work together cooperatively. This review highlights that addressing organisational and cognitive factors to help scaffold group mathematical problem solving is necessary but not sufficient. Successful group problem solving also needs to incorporate metacognitive factors in order for groups to reflect on the organisational and cognitive factors influencing their group mathematical problem solving.

Mathematical problem solving, modelling, applications, and links to other subjects

The paper will consist of three parts. In part I we shall present some background considerations which are necessary as a basis for what follows. We shall try to clarify some basic concepts and notions, and we shall collect the most important arguments (and related goals) in favour of problem solving, modelling and applications to other subjects in mathematics instruction. In the main part II we shall review the present state, recent trends, and prospective lines of development, both in empirical or theoretical research and in the practice of mathematics instruction and mathematics education, concerning problem solving, modelling, applications and relations to other subjects. In particular, we shall identify and discuss four major trends: a widened spectrum of arguments, an increased globality, an increased unification, and an extended use of computers. In the final part III we shall comment upon some important issues and problems related to our topic.

Applied mathematical problem solving, modelling, applications, and links to other subjects

The paper will consist of three parts. In part I we shall present some background considerations which are necessary as a basis for what follows. We shall try to clarify some basic concepts and notions, and we shall collect the most important arguments (and related goals) in favour of problem solving, modelling and applications to other subjects in mathematics instruction. In the main part II we shall review the present state, recent trends, and prospective lines of development, both in empirical or theoretical research and in the practice of mathematics instruction and mathematics education, concerning (applied) problem solving, modelling, applications and relations to other subjects. In particular, we shall identify and discuss four major trends: a widened spectrum of arguments, an increased globality, an increased unification, and an extended use of computers. In the final part III we shall comment upon some important issues and problems related to our topic.

Spatial thinking processes employed by primary school students engaged in mathematical problem solving

This thesis describes changes in the spatial thinking of Year 2 and Year 4 students who participated in a six-week long spatio-mathematical program. The main investigation, which contained quantitative and qualitative components, was designed to answer questions which were identified in a comprehensive review of pertinent literatures dealing with (a) young children's development of spatial concepts and skills, (b) how students solve problems and learn in different types of classrooms, and (c) the special roles of visual imagery, equipment, and classroom discourse in spatial problem solving. The quantitative investigation into the effects of a two-dimensional spatial program used a matched-group experimental design. Parallel forms of a specially developed spatio-mathematical group test were administered on three occasions—before, immediately after, and six to eight weeks after the spatial program. The test contained items requiring spatial thinking about two-dimensional space and other items requiring transfer to thinking about three-dimensional space. The results of the experimental group were compared with those of a ‘control’ group who were involved in number problem-solving activities. The investigation took into account gender and year at school. In addition, the effects of different classroom organisations on spatial thinking were investigated~one group worked mainly individually and the other group in small cooperative groups. The study found that improvements in scores on the delayed posttest of two-dimensional spatial thinking by students who were engaged in the spatial learning experiences were statistically significantly greater than those of the control group when pretest scores were used as covariates. Gender was the only variable to show an effect on the three-dimensional delayed posttest. The study also attempted to explain how improvements in, spatial thinking occurred. The qualitative component of the study involved students in different contexts. Students were video-taped as they worked, and much observational and interview data were obtained and analysed to develop categories which were described and inter-related in a model of children's responsiveness to spatial problem-solving experiences. The model and the details of children's thinking were related to literatures on visual imagery, selective attention, representation, and concept construction.

Tasks and pedagogies that facilitate mathematical problem solving

This is a report from one aspect of a project seeking to identify teacher actions that support mathematical problem solving. The project developed a planning and teaching model that describes the type of classroom tasks that can facilitate mathematical problem solving, the sequencing of the tasks, the nature of teaching heterogeneous groups, ways of dfferentiating tasks, and particular pedagogies. We report here one teacher's implementation of the
model using a unit of work that he planned and taught. The report provides important insights into the implementation of the theoretically founded model and the responses of students. We found that the model can be used for planning and teaching and for encouraging problem solving. The model has a positive effect on the learning of most students. Specific teachers actions were identified in order to address the needs of the students we are most
keen to support, those experiencing dfficulties.

Effectiveness of an Online Social Constructivist Mathematical Problem Solving Course for Malaysian Pre-Service Teachers

This study assessed the effectiveness of an online mathematical problem solving course designed using a social constructivist approach for pre-service teachers. Thirty-seven pre-service teachers at the Batu Lintang Teacher Institute, Sarawak, Malaysia were randomly selected to participate in the study. The participants were required to complete the course online without the typical face-to-face classes and they were also required to solve authentic mathematical problems in small groups of 4-5 participants based on the Polya’s Problem Solving Model via asynchronous online discussions. Quantitative and qualitative methods such as questionnaires and interviews were used to evaluate the effects of the online learning course. Findings showed that a majority of the participants were satisfied with their learning experiences in the course. There were no significant changes in the participants’ attitudes toward mathematics, while the participants’ skills in problem solving for “understand the problem” and “devise a plan” steps based on the Polya’s Model were significantly enhanced, though no improvement was apparent for “carry out the plan” and “review”. The results also showed that there were significant improvements in the participants’ critical thinking skills. Furthermore, participants with higher initial computer skills were also found to show higher performance in mathematical problem solving as compared to those with lower computer skills. However, there were no significant differences in the participants’ achievements in the course based on gender. Generally, the online social constructivist mathematical problem solving course is beneficial to the participants and ought to be given the attention it deserves as an alternative to traditional classes. Nonetheless, careful considerations need to be made in the designing and implementing of online courses to minimize problems that participants might encounter while participating in such courses.

Towards a web-based progressive handwriting recognition environment for mathematical problem solving

The emergence of pen-based mobile devices such as PDAs and tablet PCs provides a new way to input mathematical expressions to computer by using handwriting which is much more natural and efficient for entering mathematics. This paper proposes a web-based handwriting mathematics system, called WebMath, for supporting mathematical problem solving. The proposed WebMath system is based on client-server architecture. It comprises four major components: a standard web server, handwriting mathematical expression editor, computation engine and web browser with Ajax-based communicator. The handwriting mathematical expression editor adopts a progressive recognition approach for dynamic recognition of handwritten mathematical expressions. The computation engine supports mathematical functions such as algebraic simplification and factorization, and integration and differentiation. The web browser provides a user-friendly interface for accessing the system using advanced Ajax-based communication. In this paper, we describe the different components of the WebMath system and its performance analysis.

Students’ Self-Report of Motivational Orientation and Teacher Evaluation on Coping and Motivational Orientation Related to Elementary Students’ Mathematical Problem Solving and Reading Comprehension

The present study examined the correlations between motivational orientation and students’ academic performance in mathematical problem solving and reading comprehension. The main purpose is to see if students’ intrinsic motivation is related to their actual performance in different subject areas, math and reading. In addition, two different informants, students and teachers, were adopted to check whether the correlation is different by different informants. Pearson’s correlational analysis was a major method, coupled with regression analysis. The result confirmed the significant positive correlation between students’ academic performance and students’ self-report and teacher evaluation on their motivational orientation respectively. Teacher evaluation turned out with more predictive value for the academic achievement in math and reading. Between the subjects, mathematical problem solving showed higher correlation with most of the motivational subscales than reading comprehension did. The highest correlation was found between teacher evaluation on task orientation and students’ mathematical problem solving. The positive relationship between intrinsic motivation and academic achievement was proved. The disparity between students ’ self-report and teacher evaluation on motivational orientation was also addressed with the need of further examination.

Future directions and perspectives for problem solving research and curriculum development

Since the 1960s, numerous studies on problem solving have revealed the complexity of the domain and the difficulty in translating research findings into practice. The literature suggests that the impact of problem solving research on the mathematics curriculum has been limited. Furthermore, our accumulation of knowledge on the teaching of problem solving is lagging. In this first discussion paper we initially present a sketch of 50 years of research on mathematical problem solving. We then consider some factors that have held back problem solving research over the past decades and offer some directions for how we might advance the field. We stress the urgent need to take into account the nature of problem solving in various arenas of today’s world and to accordingly modernize our perspectives on the teaching and learning of problem solving and of mathematical content through problem solving. Substantive theory development is also long overdue—we show how new perspectives on the development of problem solving expertise can contribute to theory development in guiding the design of worthwhile learning activities. In particular, we explore a models and modeling perspective as an alternative to existing views on problem solving.

Methodologies for investigating relationships between concept development and the development of problem solving abilities

This paper is the second in a pair that Lesh, English, and Fennewald will be presenting at ICME TSG 19 on Problem Solving in Mathematics Education. The first paper describes three shortcomings of past research on mathematical problem solving. The first shortcoming can be seen in the fact that knowledge has not accumulated – in fact it has atrophied significantly during the past decade. Unsuccessful theories continue to be recycled and embellished. One reason for this is that researchers generally have failed to develop research tools needed to reliably observe, document, and assess the development of concepts and abilities that they claim to be important. The second shortcoming is that existing theories and research have failed to make it clear how concept development (or the development of basic skills) is related to the development of problem solving abilities – especially when attention is shifted beyond word problems found in school to the kind of problems found outside of school, where the requisite skills and even the questions to be asked might not be known in advance. The third shortcoming has to do with inherent weaknesses in observational studies and teaching experiments – and the assumption that a single grand theory should be able to describe all of the conceptual systems, instructional systems, and assessment systems that strongly molded and shaped by the same theoretical perspectives that are being used to develop them. Therefore, this paper will describe theoretical perspectives and methodological tools that are proving to be effective to combat the preceding kinds or shortcomings. We refer to our theoretical framework as models & modeling perspectives (MMP) on problem solving (Lesh & Doerr, 2003), learning, and teaching. One of the main methodologies of MMP is called multi-tier design studies (MTD).

Problem solving for the 21st century

Mathematical problem solving has been the subject of substantial and often controversial research for several decades. We use the term, problem solving, here in a broad sense to cover a range of activities that challenge and extend one’s thinking. In this chapter, we initially present a sketch of past decades of research on mathematical problem solving and its impact on the mathematics curriculum. We then consider some of the factors that have limited previous research on problem solving. In the remainder of the chapter we address some ways in which we might advance the fields of problem-solving research and curriculum development.

Problem solving in a 21st century mathematics curriculum

Research on problem solving in the mathematics curriculum has spanned many decades, yielding pendulum-like swings in recommendations on various issues. Ongoing debates concern the effectiveness of teaching general strategies and heuristics, the role of mathematical content (as the means versus the learning goal of problem solving), the role of context, and the proper emphasis on the social and affective dimensions of problem solving (e.g., Lesh & Zawojewski, 2007; Lester, 2013; Lester & Kehle, 2003; Schoenfeld, 1985, 2008; Silver, 1985). Various scholarly perspectives—including cognitive and behavioral science, neuroscience, the discipline of mathematics, educational philosophy, and sociocultural stances—have informed these debates, often generating divergent resolutions. Perhaps due to this uncertainty, educators’ efforts over the years to improve students’ mathematical problem-solving skills have had disappointing results. Qualitative and quantitative studies consistently reveal mathematics students’ struggles to solve problems more significant than routine exercises (OECD, 2014; Boaler, 2009)...