Dissecting Foundation Thinking About Mathematics - Sept. 2016 (1).doc

Proceedings of the Thirty-first Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, edited by S. L. Swars, D. W. Stinson, & S. Lemons-Smith. (pp. 1576-1583). Atlanta: Georgia State University, 2009., 2009

The following is the abstract (as submitted). It was a description of the proposed session and a call for participants. The idea of “mathematical habits of mind” has been introduced to emphasize the need to help students think about mathematics “the way mathematicians do.” There seems to be considerable interest among mathematics educators and mathematicians in helping students develop mathematical habits of mind. The objectives of this working group are: (a) to discuss various views and aspects of mathematical habits of mind, (b) to explore avenues for research, (c) to encourage research collaborations, and (d) to interest doctoral students in this topic. To facilitate the discussion during the working group meetings, we provide an overview of mathematical habits of mind, including concepts that are closely related to habits of mind—ways of thinking, mathematical practices, knowing-to act in the moment, cognitive disposition, and behavioral schemas. We invite mathematics educators who are interested in habits of mind, and especially those who have conducted research related to habits of mind, to share their work during the first working group meeting. If you would like to give a 10-minute presentation, please contact Kien Lim or Annie Selden in advance.

Mathematical Thinking and Learning, 7(1), 2005, 1-13., 2005

This article sets the stage for the following 3 articles. It opens with a brief history of attempts to characterize advanced mathematical thinking, beginning with the deliberations of the Advanced Mathematical Thinking Working Group of the International Group for the Psychology of Mathematics Education. It then locates the articles within 4 recurring themes: (a) the distinction between identifying kinds of thinking that might be regarded as advanced at any grade level, and taking as advanced any thinking about mathematical topics considered advanced; (b) the utility of characterizing such thinking for integrating the entire curriculum; (c) general tests, or criteria, for identifying advanced mathematical thinking; and (d) an emphasis on advancing mathematical practices. Finally, it points out some commonalities and differences among the 3 following articles.

Journal for Research in Mathematics Education, 2012

Imagine you have woken from a dream, a dream brimming with meaning, with passion, with mystery. You try to sustain the feeling, recount the details, share the experience. You fail. Your powers of reconstruction too meager, your tongue too clumsy. Mathematics is such a dream, dreamed by individuals, personal, yet remarkably in a waking state, and provoking sufficient commonality in its recounting to bring individuals together, to create a community of shared passion. The first dreamer of the dream, shrouded in history and myth, perhaps was Thales of Miletus, who later advised Pythagoras of Samos who subsequently founded an order of adherents holding knowledge and property in common while pursuing philosophical and mathematical studies as a moral basis for the conduct of life

Research into mathematics often focuses on basic numerical and spatial intuitions, and one key property of numbers: their magnitude. The fact that mathematics is a system of complex relationships that invokes reasoning usually receives less attention. The purpose of this special issue is to highlight the intricate connections between reasoning and mathematics, and to use insights from the reasoning literature to obtain a more complete understanding of the processes that underlie mathematical cognition. The topics that are discussed range from the basic heuristics and biases to the various ways in which complex, effortful reasoning contributes to mathematical cognition, while also considering the role of individual differences in mathematics performance. These investigations are not only important at a theoretical level, but they also have broad and important practical implications, including the possibility to improve classroom practices and educational outcomes, to facilitate people's decision-making, as well as the clear and accessible communication of numerical information.

2011

Some years ago 1 , in what was a discussion of other matters, we wondered about the analytic potentiality of ignorance (Sharrock and Anderson, 1980). What had caught our eye was the possibility, no more, that the initial fieldwork experience, involving as it does the overcoming of alienness and separation, could provide rich resources for the analysis of culture. In coming to know a culture, in finding one’s feet, the fieldworker finds an organisation to activities, knowledge and practices. That organisation constitutes the culture for him. In focussing on this “discovery”, it might be possible to draw out some of the ways that bodies of knowledge and associated activities make themselves accessible, comprehensible and utilisable by anyone who comes to them. Such a suggestion is, of course, little more than an extension of Schutz’ (1962) insights with

What is mathematics? What does it mean to be a mathematician? What should students understand about the nature of mathematical knowledge and inquiry? Research in the field of mathematics education has found that students often have naïve views about the nature of mathematics. Some believe that mathematics is a body of unchanging knowledge, a collection of arbitrary rules and procedures that must be memorized. Mathematics is seen as an impersonal and uncreative subject. To combat the naïve view, we need a humanistic vision and explicit goals for what we hope students understand about the nature of mathematics. The goal of this dissertation was to begin a systematic inquiry into the nature of mathematics by identifying humanistic characteristics of mathematics that may serve as goals for student understanding, and to tell real-life stories to illuminate those characteristics. Using the methodological framework of heuristic inquiry, the researcher identified such characteristics by collaborating with a professional mathematician, by co-teaching an undergraduate transition-to-proof course, and being open to mathematics wherever it appeared in life. The results of this study are the IDEA Framework for the Nature of Pure Mathematics and ten corresponding stories that illuminate the characteristics of the framework. The IDEA framework consists of four foundational characteristics: Our mathematical ideas and practices are part of our Identity; mathematical ideas and knowledge are Dynamic and forever refined; mathematical inquiry is an emotional Exploration of ideas; and mathematical ideas and knowledge are socially vetted through Argumentation. The stories that are told to illustrate the IDEA framework capture various experiences of the researcher, from conversations with his son to emotional classroom discussions between undergraduates in a transition-to-proof course. The researcher draws several implications for teaching and research. He argues that the IDEA framework should be tested in future research for its effectiveness as an aid in designing instruction that fosters humanistic conceptions of the nature of mathematics in the minds of students. He calls for a cultural renewal of undergraduate mathematics instruction, and he questions the focus on logic and set theory within transition-to-proof courses. Some instructional alternatives are presented. The final recommendation is that nature of mathematics become a subject in its own right for both students and teachers. If students and teachers are to revise their beliefs about the nature of mathematics, then they must have the opportunities to reflect on what they believe about mathematics and be confronted with experiences that challenge those beliefs.

This article sheds light on the single phrase, logical thinking, which came to be understood in so many diverse ways. To assist explain the many distinct meanings, how they arose, and how they are connected, we trace the emergence and evolution of logical thinking in mathematics. This article is also, to some extent, a description of a movement that arose outside of philosophy's mainstream, and whose beginnings lay in a desire to make logic practical and an essential part of learners' lives.

CMESG/GCEDM Proceedings 2003, 2003

Historical development is sometimes used as a format for teaching mathematics in school. However, there is a significant difference between the historical development of sophisticated adults in successive societies and the development of the child in today's society. This talk will formulate a theoretical framework based on genetic facilities we all share that are the basis of mathematical thinking and operate in history and the experiences that we have in life that lead to the personal development of mathematical thinking. The genetic facilities that are set-before our birth include three major facilities: language, recognition of similarities and differences and repetition of a sequence of actions until they can be performed routinely. Language enables us to categorise things, as we do in geometry, and to name actions such as counting to be symbolised as number and algebra. We repeat actions again and again to get the potential infinity of whole numbers and we categorise the whole numbers as a set leading to actual infinity. The three set-befores of recognition, repetition and language lead to three distinct worlds of mathematical thinking: embodied (based on our perceptions and actions), symbolic (based on the symbolisation of actions as thinkable concepts) and formal (based on set-theoretic definitions and formal proofs). As children learn, they build on their previous experiences, met-before in their life to build individual meanings for mathematics. These met-befores act in part to build new concepts, but can also be inappropriate when used in new contexts. The notions of set-before and met-before within three distinct worlds of mathematics have vital consequences in the interpretation of history and in the teaching of mathematics.