The goals of this chapter are (1) to outline and substantiate a broad conceptualization of what it means to think mathematically, (2) to summarize the literature relevant to understanding mathematical thinking and problem solving, and (3) to point to new directions in research, development, and assessment consonant with an emerging understanding of mathematical thinking and the goals for instruction outlined here. The use of the phrase “learning to think mathematically” in this chapter’s title is deliberately broad. Although the original charter for this chapter was to review the literature on problem solving and metacognition, the literature itself is somewhat ill defined and poorly grounded. As the literature summary will make clear, problem solving has been used with multiple meanings that range from “working rote exercises” to “doing mathematics as a professional”; metacognition has multiple and almost disjoint meanings (from knowledge about one’s thought processes to self-regulation during problem solving) that make it difficult to use as a concept. This chapter outlines the various meanings that have been ascribed to these terms and discusses their role in mathematical thinking. The discussion will not have the character of a classic literature review, which is typically encyclopedic in its references and telegraphic in its discussions of individual papers or results. It will, instead, be selective and illustrative, with main points illustrated by extended discussions of pertinent examples. Problem solving has, as predicted in the 1980 Yearbook of the National Council of Teachers of Mathematics (Krulik, 1980, p. xiv), been the theme of the 1980s. The decade began with NCTM’s widely heralded statement, in its Agenda for Action, that “problem solving must be the focus of school mathematics” (NCTM, 1980, p. 1). It concluded with the publication of Everybody Counts (National Research Council, 1989) and the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), both of which emphasize problem solving. One might infer, then, that there is general acceptance of the idea that the primary goal of mathematics instruction should be to have students become competent problem solvers. Yet, given the multiple interpretations of the term, the goal is hardly clear. Equally unclear is the role that problem solving, once adequately characterized, should play in the larger context of school mathematics. What are the goals for mathematics instruction, and how does problem solving fit within those goals? Such questions are complex. Goals for mathematics instruction depend on one’s conceptualization of what mathematics is, and what it means to understand mathematics. Such conceptualizations vary widely. At one end of the spectrum, mathematical knowledge is seen as a body of facts and procedures dealing with quantities, magnitudes, and forms, and the relationships among them; knowing mathematics is seen as having mastered these facts and procedures. At the other end of the spectrum, mathematics is conceptualized as the “science of patterns,” an (almost) empirical discipline closely akin to the sciences in its emphasis on pattern-seeking on the basis of empirical evidence. The author’s view is that the former perspective trivializes mathematics; that a curriculum based on mastering a corpus of mathematical facts and procedures is severely impoverished—in much the same way that an English curriculum would be considered impoverished if it focused largely, if not exclusively, on issues of grammar. The author characterizes the mathematical enterprise as follows:

Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics (Reprint):

Constructivist theory has been prominent in recent research on mathematics learning and has provided a basis for recent mathematics education reform efforts. Although constructivism has the potential to inform changes in mathematics teaching, it offers no particular vision of how mathematics should be taught; models of teaching based on constructivism are needed. Data are presented from a whole-class, constructivist teaching experiment in which problems of teaching practice required the teacher/researcher to explore the pedagogical implications of his theoretical (constructivist) perspectives. The analysis of the data led to the development of a model of teacher decision making with respect to mathematical tasks. Central to this model is the creative tension between the teacher's goals with regard to student learning and his responsibility to be sensitive and responsive to the mathematical thinking of the students.

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Reconstructing Mathematics Pedagogy from a Constructivist Perspective.

ion is not inevitable. In fact, shopkeepers have never abstracted the structures of modules which regulate their financial exchanges because they haven’t the motivation to do so. Schematization and formulation The schematization and the formulation of the structure follows the process of identification and of updating. Diénès does not plan any general situations specific to this stage (would something which is well conceived of spell itself out clearly?) but the representation by a graph is often envisaged as a simplified but natural and direct expression of the thought of a child. Representing objects by points and operators by arrows is learned by the use of imitation, like a language. Symbolization Symbolization is the transcription in a new language of properties represented in the preceding stage.

https://link.springer.com/content/pdf/bfm%3A978-0-306-47211-4%2F1.pdf

Theory of didactical situations in mathematics

Mobile devices such as laptops, personal digital assistants, and mobile phones have become a learning tool with great potential in both classrooms and outdoor learning. Although there have been qualitative analyses of the use of mobile devices in education, systematic quantitative analyses of the effects of mobile-integrated education are lacking. This study performed a meta-analysis and research synthesis of the effects of integrated mobile devices in teaching and learning, in which 110 experimental and quasiexperimental journal articles published during the period 1993-2013 were coded and analyzed. Overall, there was a moderate mean effect size of 0.523 for the application of mobile devices to education. The effect sizes of moderator variables were analyzed and the advantages and disadvantages of mobile learning in different levels of moderator variables were synthesized based on content analyses of individual studies. The results of this study and their implications for both research and practice are discussed. This is a meta-analysis and research synthesis study for mobile-integrated education.110 published journal articles that were written over a 20-year period were coded and analyzed.The application of mobile devices to education has a moderate mean effect size.The effect sizes of moderator variables were analyzed.The benefits and drawbacks of mobile learning were synthesized.

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The effects of integrating mobile devices with teaching and learning on students' learning performance

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Millennials Rising: The Next Great Generation

Over the next 10 years, we anticipate that personal, portable, wirelessly-networked technologies will become ubiquitous in the lives of learners - indeed, in many countries, this is already a reality. We see that ready-to-hand access creates the potential for a new phase in the evolution of technology-enhanced learning (TEL), characterized by "seamless learning spaces" and marked by continuity of the learning experience across different scenarios (or environments), and emerging from the availability of one device or more per student ("one-to-one"). One-to-one TEL has the potential to "cross the chasm" from early adopters conducting isolated design studies to adoption-based research and widespread implementation, with the help of research and evaluation that gives attention to the digital divide and other potentially negative consequences of pervasive computing. We describe technology-enhanced learning and the affordances of one-to-one computing and outline a research agenda, including the risks and challenges of reaching scale. We reflect upon how this compares with prior patterns of technology innovation and diffusion. We also introduce a community, called "G1:1," that brings together leaders of major research laboratories and one-to-one TEL projects. We share a vision of global research, inviting other research groups to collaborate in ongoing activities.

/pdf/one-to-one-technology-enhanced-learning-an-opportunity-for-xawanamjqn.pdf

One-to-one technology-enhanced learning: An opportunity for global research collaboration

Nous etudions les relations entre preuves et contradictions dans la resolution d'un probleme de mathematiques. Cette etude montre la necessite d'une approche a la fois situationnelle et cognitive, notamment en reference au fonctionnement des connaissances dans l'apprentissage des mathematiques. Ceci nous conduit a distinguer differents stades dans l'evolution des preuves pragmatiques aux preuves intellectuelles. Enfin nous montrons que le depassement d'une contradiction ne constitue pas necessairement un progres cognitif, en particulier nous examinons le traitement d'un contre-exemple par des eleves de quatrieme.

https://hal.archives-ouvertes.fr/hal-01619264/document

Processus de preuve et situations de validation

Technology-enhanced learning is a timely topic, the importance of which is recognized by educational researchers, practitioners, software designers, and policy makers. This volume presents and discusses current trends and issues in technology-enhanced learning from a European research and development perspective. This multifaceted and multidisciplinary topic is considered from four different viewpoints, each of which constitutes a separate section in the book. The sections include general as well as domain-specific principles of learning that have been found to play a significant role in technology-enhanced environments, ways to shape the environment to optimize learners interactions and learning, and specific technologies used by the environment to empower learners. An additional section discusses the work presented in the preceding sections from a computer science perspective and an implementation perspective. This book comes out of the work in Kaleidoscope: a European Network of Excellence in which over 1,000 people from more than 90 institutes across Europe participate. Kaleidoscope brings together researchers from diverse disciplines and cultures, through their collaboration and sharing of scientific outcomes, they are helping move the field of technology-enhanced learning forward.

Technology-Enhanced Learning: Principles and Products

This chapter attempts to set a perspective on where interactive technologies have taken us and where they seem to be headed. After briefly reviewing their impact in different mathematical domains, including arithmetic, algebra, geometry, statistics, and calculus, we examine what we believe to be the sources of technology’s power, which we feel is primarily epistemological. While technology’s impact on daily practice has yet to match expectations from two or three decades ago, its epistemological impact is deeper than expected. This impact is based in a reification of mathematical objects and relations that students can use to act more directly on these objects and relations than ever before. This new mathematical realism, when coupled with the fact that the computer becomes a new partner in the didactical contract, forces us to extend the didactical transposition of mathematics to a computational transposition. This new realism also drives ever deeper changes in the curriculum, and it challenges widely held assumptions about what mathematics is learnable by which students, and when they may learn it. We also examine the limits of Artificial Intelligence and microworlds and how these may be changing. We close by considering the newer possibilities offered by the Internet and its dramatic impact on connections among learners, teachers, and the immense resources that are becoming available to both. Our conclusion is that we are very early in the technological transformation and that we desperately need research in all aspects of teaching and learning with technology.

Computer-Based Learning Environments in Mathematics

This article presents the main features of the theoretical framework of French research known as recherches en didactique des mathematiques. The foundation of this approach consists mainly of the relationships between two hypotheses and two constraints, which are presented together with some specific key words. Outlines are given of Brousseau's thdorie des situations didactiques (theory of didactical situations). An example is given that presents in some detail the rationale for the construction of a didactical situation and its analysis. This article ends with some questions addressed to research on mathematics teaching.

Nicolas Balacheff

Papers

One-to-one technology-enhanced learning: An opportunity for global research collaboration

Processus de preuve et situations de validation

Technology-Enhanced Learning: Principles and Products

Computer-Based Learning Environments in Mathematics

Towards a problematique for research on mathematics teaching