Mathematical Cognition

The conjunction fallacy is the well-documented reasoning error on which people rate a conjunction A∧B as more probable than one of its conjuncts, A. Many explanations appeal to the fact that B has a high probability in the given scenarios, but Katya Tentori and collaborators have challenged such approaches. They report experiments suggesting that degree of confirmation—rather than probability—is the central determinant of the conjunction fallacy. In this paper, we have two goals. First, we address a confound in Tentori et (...) al.’s experiments: they failed to control for the fact that in their stimuli where B is confirmed, it is also conversationally relevant in the sense that it fits with the topic or question under discussion. Conversely, when B has a high probability but is not confirmed, it is conversationally irrelevant. Consequently, it is possible that conversational relevance, rather than confirmation, is responsible for the differences they found between confirmed and probable hypotheses. Second, inspired by recent theoretical work, we aim to give the first empirical investigation of the hypothesis that this type of conversational relevance on its own—independently of degree of confirmation—can be an important factor in the conjunction fallacy. We report on two experiments that vary Tentori et al.’s design by making B relevant without changing its degree of probability or confirmation. We found that doing so increases the rate of the conjunction fallacy, suggesting that relevance plays an important role in the conjunction fallacy. (shrink)

Questions regarding the nature and acquisition of mathematical knowledge are perhaps as old as mathematical thinking itself, while fundamental issues of mathematical ontology and epistemology have direct bearing on mathematical cognition. Several original contributions to logic and mathematics made by the American polymath, Charles Sanders Peirce, are of direct relevance to these fundamental issues. This chapter explores scientific reasoning as it relates to abduction, a name that Peirce coined for educated “guessing” of hypotheses, which he took to be “the first (...) step of scientific reasoning” and the only creative one. Yet he also argued that all deductive reasoning is mathematical and that all mathematical reasoning is diagrammatic. Representation, especially in the form of a diagrammatic system of logic that Peirce developed, is explored here along with his logic of inquiry, most notably in terms of its manifestation as the logic of ingenuity. Originating in the field of engineering, here the diagram of a problem serves as a heuristic substitute for evaluating the actual situation, an approach that can be extended to other forms of practical reasoning such as ethical deliberation. This chapter also touches upon such diverse but related subjects as non-Euclidean geometry and nonclassical logic, with additional examples that help to elucidate cognitive elements of mathematical knowledge. (shrink)

Cognitive mathematics provides insights into how mathematics works inside the brain and how it is interconnected with other faculties through so-called blending and other associative processes. This handbook is the first large collection of various aspects of cognitive mathematics to be amassed into a single title, covering decades of connection between mathematics and other figurative processes as they manifest themselves in language, art, and even algorithms. It will be of use to anyone working in math cognition and education, with each (...) section of the handbook edited by an international leader in that field. (shrink)

By analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are imagined objects, some of which, at least approximately, exist in our internal world of activities or we can realize (...) or represent them there; (ii) mathematical truths are not truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. (shrink)

Number Nativism is the view that humans innately represent precise natural numbers. Despite a long and venerable history, it is often considered hopelessly out of touch with the empirical record. I argue that this is a mistake. After clarifying Number Nativism and distancing it from related conjectures, I distinguish three arguments which have been seen to refute the view. I argue that, while popular, two of these arguments miss the mark, and fail to place pressure on Number Nativism. Meanwhile, a (...) third argument is best construed as a challenge: rather than refuting Number Nativism, it challenges its proponents to provide positive evidence for their thesis and show that this can be squared with apparent counterevidence. In response, I introduce psycholinguistic work on The Tolerance Principle (not yet considered in this context), propose that it is hard to make sense of without positing precise and innate representations of natural numbers, and argue that there is no obvious reason why these innate representations couldn’t serve as a basis for mature numeric conception. (shrink)

Does the visual system adapt to number? For more than fifteen years, most have assumed that the answer is an unambiguous “yes”. Against this prevailing orthodoxy, we recently took a critical look at the phenomenon, questioning its existence on both empirical and theoretical grounds, and providing an alternative explanation for extant results (the old news hypothesis). We subsequently received two critical responses. Burr, Anobile, and Arrighi rejected our critiques wholesale, arguing that the evidence for number adaptation remains overwhelming. Durgin questioned (...) our old news hypothesis — preferring instead a theory about density adaptation he has championed for decades — but also highlighted several ways in which our arguments do pose serious challenges for proponents of number adaptation. Here, we reply to both. We first clarify our position regarding number adaptation. Then, we respond to our critics’ concerns, highlighting seven reasons why we remain skeptical about number adaptation. We conclude with some thoughts about where the debate may head from here. (shrink)

How do we acquire the notions of cardinality and cardinal number? In the (neo-)Fregean approach, they are derived from the notion of equinumerosity. According to some alternative approaches, defended and developed by Husserl and Parsons among others, the order of explanation is reversed: equinumerosity is explained in terms of cardinality, which, in turn, is explained in terms of our ordinary practices of counting. In their paper, ‘Cardinality, Counting, and Equinumerosity’, Richard Kimberly Heck proposes that instead of equinumerosity or counting, cardinality (...) is derived from a cognitively earlier notion of _just as many_. In this paper, we assess Heck’s proposal in terms of contemporary theories of number concept acquisition. Focusing on bootstrapping theories, we argue that there is no evidence that the notion of _just as many_ is cognitively primary. Furthermore, since the acquisition of cardinality is an enculturated process, the cognitive primariness of these notions, possibly including _just as many_, depends on various external cultural factors. Therefore, being possibly a cultural construction, _just as many_ could be one among several notions used in the acquisition of cardinality and cardinal number concepts. This paper thus challenges those accounts which seek for a fundamental concept underlying all aspects of numerical cognition. (shrink)

The Cognitive Foundations and Epistemology of Arithmetic and Geometry How is knowledge of arithmetic and geometry developed and acquired? In the tradition established by Plato and often associated with Kant, the epistemology of mathematics has been focused on a priori approaches, which take mathematical knowledge and its study to be essentially independent of sensory experience. … Continue reading The Cognitive Foundations and Epistemology of Arithmetic and Geometry →.

Arithmetic is one of the foundations of our educational systems, but what exactly is it? Numbers are everywhere in our modern societies, but what is our knowledge of numbers really about? This book provides a philosophical account of arithmetical knowledge that is based on the state-of-the-art empirical studies of numerical cognition. It explains how humans have developed arithmetic from humble origins to its modern status as an almost universally possessed knowledge and skill. Central to the account is the realisation that, (...) while arithmetic is a human creation, the development of arithmetic is constrained by our evolutionarily developed cognitive architecture. Arithmetic is a sophisticated cultural development, but it is ultimately based on abilities with numerosities that we already possess as infants and share with many non-human animals. Therefore, arithmetic is not purely conventional, an arbitrary game akin to chess. Instead, arithmetic is deeply connected to our basic cognitive capacities. (shrink)

The goal of this unique text is to provide an "experience" that would facilitate a better transition for mathematics majors to the advanced proof-based courses required for their major. If you "love mathematics, but I hate proofs" this book is for you. Example-based courses such as introductory Calculus transition somewhat abruptly, and without a warning label, to proof-based courses, and may leave students with the unpleasant feeling that a subject they loved has turned into material they find hard to understand. (...) The book exposes students and readers to the fundamental nature and principles of constructing mathematical proofs and in the context of main courses required for the major, e.g., probability, linear algebra, real analysis, and abstract algebra. Four short chapters, each chapter focusing on a particular course, provide a short but rigorous introduction. Students then get a preview of the discipline, its focus, language, mathematical objects of interests, and common methods of proof presented in those courses. Because which ideas apply to which future courses may not be obvious in many transition courses, this structure addresses this need. The book may also be used as a review tool at the end of course and for readers who want to learn the language and scope of the broad disciplines of linear algebra, abstract algebra, real analysis, and probability, before transitioning to these courses. (shrink)

This paper considers hybrid systems — dynamical systems that exhibit both continuous and discrete behavior. Usually, in these systems, interactions between the continuous and discrete dynamics occur when a pre-defined function becomes equal to zero, i.e., in the system occurs a zero-crossing (the situation where the function only “touches” zero is considered as the zero-crossing, as well). Determination of zero-crossings plays a crucial role in the correct simulation of the system in this case. However, for models of many real-life hybrid (...) systems, such interactions may lead to the so-called Zeno executions, i.e., situations where the system undergoes an unbounded number of discrete transitions in a finite and bounded length of time. In this case, standard numerical methods of simulating the systems may fail, since the time between two transitions can decrease significantly leading to ill-conditioning of the simulation. Correct determination of zero-crossings for a complex real-life system can require a lot of computational resources and, as a consequence, slow down the simulation significantly. This paper presents a new way to execute the simulation generating time observations of the hybrid system dynamically using numerical infinitesimals introduced recently, allowing thus to determine zero-crossings more accurately. The proposed method allows to automatically detect zero-crossings with predefined accuracy and to analyze better the behavior of the system around the zero-crossings generating observations more densely, where it is necessary. Moreover, the search for zero-crossings is performed efficiently without re-evaluation of the whole system at each observation. To show the validity of the proposed algorithm, the well-known Bouncing Ball hybrid system has been studied and the obtained simulation results were compared with the standard method. (shrink)

Sand drawings are introduced in relation to the fieldwork of British anthropologists John Layard and Bernard Deacon early in the twentieth century, and the status of sand drawings as mathematics is discussed in the light of Wittgenstein’s idea that “in mathematics process and result are equivalent”. Included are photographs of the illustrations in Layard’s own copy of Deacon’s “Geometrical Drawings from Malekula and other Islands of the New Hebrides” (1934). This is a brief companion to my article “Wittgenstein on string (...) figures as mathematics” (2022), published in the journal Philosophical Investigations. (shrink)

In this paper, I study the kind of questions we can ask about the existence of numbers. In addition to asking whether numbers exist, and how, I argue that there is also a third relevant question: why numbers exist. In platonist and nominalist accounts this question may not make sense, but in the psychologist account I develop, it is as well-placed as the other two questions. In fact, there are two such why-questions: the causal why-question asks what causes numbers to (...) exist and the teleological why-question asks for what purpose numbers exist. I argue that in a psychologist constructivist account, in which numbers are understood to exist as referents of a particular type of culturally shared concepts, both why-questions can get plausible answers. (shrink)

I argue that analogue mental representations possess a canonical decomposition into privileged constituents from which they compose. I motivate this suggestion, and rebut arguments to the contrary, through reflection on the approximate number system, whose representations are widely expected to have an analogue format. I then argue that arguments for the compositionality and constituent structure of these analogue representations generalize to other analogue mental representations posited in the human mind, such as those in early vision and visual imagery.

Combining inquiry and metacognition helps strengthen mathematical learning. This study examines how metacognitive mathematical inquiry can be modeled using reflective tasking approach. Quasi-experimental design was employed in two comparable groups of Grade 9 students of Ibarra National High School, Maasin City, Philippines during the academic year 2021-2022. Lesson guides on reflective task assessments anchored on metacognitive and inquiry-based learning theories, inquiry rubric scales and modified state metacognitive inventory served as data collection instruments. Results of t-test analysis revealed significant difference in (...) performance between groups implying high efficacy of the methodology in stimulating students’ metacognitive inquiry skill. Correlation values projected significant relationship between reflective learning performances with metacognition. Quantitative evidences, thus, lead to conclusion that reflective tasking methodology exemplifies potential in metacognitive mathematical inquiry as it optimizes their thinking processes, task involvement, and regulation through self-reflection principles worthy of practice throughout lifelong learning. (shrink)

This paper investigates some classical oppositional categories, like synthetic vs. analytic, posterior vs. prior, imagination vs. grammar, metaphor vs. hermeneutics, metaphysics vs. observation, innovation vs. routine, and image vs. sound, and the role they play in epistemology and philosophy of science. The epistemological framework of objective cognitive constructivism is of special interest in these investigations. Oppositional relations are formally represented using algebraic lattice structures like the cube and the hexagon of opposition, with applications in the contexts of modern color theory, (...) Kantian philosophy, Jungian psychology, and linguistics. (shrink)

We propose the concepts of philomatics and psychomatics as hybrid combinations of philosophy and psychology with mathematics. We explain four motivations for this combination which are fulfilling the desire of analytical philosophy, proposing science of philosophy, justifying mathematical algorithms by philosophy, and abstraction in both philosophy and mathematics. We enumerate various examples for philomatics and psychomatics, some of which are explained in more depth. The first example is the analysis of relation between the context principle, semantic holism, and the usage (...) theory of meaning with the attention mechanism in mathematics. The other example is on the relations of Plato's theory of forms in philosophy with the holographic principle in string theory, object-oriented programming, and machine learning. Finally, the relation between Wittgenstein's family resemblance and clustering in mathematics is explained. This paper opens the door of research for combining philosophy and psychology with mathematics. (shrink)

This textbook is designed for an Introduction to Proofs course organized around the themes of number and space. Concepts are illustrated using both geometric and number examples, while frequent analogies and applications help build intuition and context in the humanities, arts, and sciences. Sophisticated mathematical ideas are introduced early and then revisited several times in a spiral structure, allowing students to progressively develop rigorous thinking. Throughout, the presentation is enlivened with whimsical illustrations, apt quotations, and glimpses of mathematical history and (...) culture. Early chapters integrate an introduction to sets, logic, and beginning proof techniques with a first exposure to more advanced mathematical structures. The middle chapters focus on equivalence relations, functions, and induction. Carefully chosen examples elucidate familiar topics, such as natural and rational numbers and angle measurements, as well as new mathematics, such as modular arithmetic and beginning graph theory. The book concludes with a thorough exploration of the cardinalities of finite and infinite sets and, in two optional chapters, brings all the topics together by constructing the real numbers and other complete metric spaces. Designed to foster the mental flexibility and rigorous thinking needed for advanced mathematics, Introduction to Mathematics suits either a lecture-based or flipped classroom. A year of mathematics, statistics, or computer science at the university level is assumed, but the main prerequisite is the willingness to engage in a new challenge. (shrink)

In this article, modern standards of early years mathematics education are criticized and a proposal for change is presented. Today's early years mathematics education standards rest on a view of mathematics that became obsolete already at the end of the 19th century while the spirit of children's mathematics is precisely the spirit of modern mathematics. The proposal for change is not a return to the “new mathematics” movement, but something different.

The study sought to identify and evaluate the classroom assessment thoughts, practices, and skills of secondary mathematics teachers in Bulacan. The study revealed that there are no significant relationships between teachers’ thoughts of classroom assessments and practices, and classroom assessment practices and skills. However, there is significant relationship between teachers’ thoughts of classroom assessments and skills. There are no significant differences between the teachers’ thoughts of classroom assessments and their age, educational attainment, teaching experience, number of years in teaching mathematics (...) and in-service training. There is a significant relationship between the teachers’ thoughts of classroom assessment and their classroom assessment skills. In the light of the findings, the researcher recommends that: Secondary mathematics teachers should adopt varied and innovative ways of assessing students’ learning in mathematics. School administrators should ensure that all teachers with appropriate certificates must take more training courses in assessment to improve their skills and use of desirable classroom assessment practices. Further research may focus on the relationship between mathematics teachers’ assessment skills and thoughts, practices and skills, and thoughts and practices. (shrink)

It has become a fact that fluency and competency in utilizing the advancement of technology, specifically the computer and the internet is one way that could help in facilitating learning in mathematics. This study investigated the effects of Dynamic Geometry Software (DGS) and Computer Algebra Systems (CAS) in teaching Mathematics. This was conducted in Zamboanga del Sur National High School (ZSNHS) during the third grading period of the school year 2015-2016. The study compared the achievement and attitude towards Mathematics between (...) those students taught with DGS and CAS and those students who were taught without DGS and CAS. A quasi-experimental pretest-posttest control group design was used with two groups which was matched according to their mathematical level. An achievement test was developed and administered to assess the students’ achievement in Mathematics. Aiken’s attitude scale was also used to assess the students’ attitude towards Mathematics. The data were treated with One-Way Analysis of Covariance (ANCOVA) and t-test at 0.05 level of significance. The study revealed a significant difference between the mathematics achievement between students taught with DGS and CAS and those who were taught without it. The integration of DGS and CAS in Mathematics lessons leads to better academic achievement result compared with that of the conventional method of teaching. Moreover, there was no significant difference on the attitude level of the students between the two groups. (shrink)

Créativité et mathématiques ne font pas bon ménage dans l'inconscient collectif. Pourtant à l'heure d'Internet, il existe bien une manière de revisiter la géométrie, l'algèbre ou même la logique. Il suffit de prendre un peu de distance par rapport aux calculs, de se donner quelques libertés par rapport à l'histoire, de changer de point de vue... En vagabondant à travers l'histoire et l'application des mathématiques, les auteurs se livrent ici à un jeu de vulgarisation d'une saveur et d'une subtilité inédites. (...) [4e de couv.]. (shrink)

It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using ①-based infinite numbers is applied to measure the set A (where the number ① is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable (...) or it has the cardinality of the continuum, the ①-based methodology can provide a more accurate measurement of infinite sets. In this article, lower and upper estimates of the number of elements of A are obtained. Both estimates are expressed in ①-based numbers. (shrink)

Numbers. Natural numbers -- Integers -- Real numbers -- Irrational and transcendental numbers -- Funny numbers. Zero -- e : the unnatural natural number -- [Phi] : the golden ratio -- i : the imaginary number -- Writing numbers. Roman numerals -- Egyptian fractions -- Continued fractions -- Logic. Mr. Spock is not logical -- Proofs, truth, and trees : oh my! -- Programming with logic -- Temporal reasoning -- Sets. Cantor's diagonalization : infinity isn't just infinity -- Axiomatic set (...) theory : keep the good, dump the bad -- Models : using sets as the LEGOs of the math world -- Transfinite numbers : counting and ordering infinite sets -- Group theory : finding symmetries with sets -- Mechanical math. Finite state machines : simplicity goes far -- The turing machine -- Pathology and the heart of computing -- Calculus : no, not that calculus- [lambda] calculus -- Numbers, booleans, and recursion -- Types, types, types : modeling [lambda] calculus -- The halting problem. (shrink)

I. Prelude -- About this Book -- II. School Mathematics and Its Consequences -- The Foundations of Mathematical Thinking -- Compression, Connection and Blending of Mathematical Ideas -- Set-befores, Met-befores and Long-term Learning -- Mathematics and the Emotions -- The Three Worlds of Mathematics -- Journeys through Embodiment and Symbolism -- Problem-Solving and Proof -- III. Interlude -- The Historical Evolution of Mathematics -- IV. University Mathematics and Beyond -- The Transition to Formal Knowledge -- Blending Knowledge Structures in the (...) Calculus -- Expert Thinking and Structure Theorems -- Contemplating the Infinitely Large and the Infinitely Small -- Expanding the Frontiers through Mathematical Research -- Reflections -- Appendix: Where the Ideas Came From. (shrink)

This book is devoted to the study of human thought, its systemic structure, and the historical development of mathematics both as a product of thought and as a fascinating case analysis. After demonstrating that systems research constitutes the second dimension of modern science, the monograph discusses the yoyo model, a recent ground-breaking development of systems research, which has brought forward revolutionary applications of systems research in various areas of the traditional disciplines, the first dimension of science. After the systemic structure (...) of thought is factually revealed, mathematics, as a product of thought, is analyzed by using the age-old concepts of actual and potential infinities. In an attempt to rebuild the system of mathematics, this volume first provides a new look at some of the most important paradoxes, which have played a crucial role in the development of mathematics, in proving what these paradoxes really entail. Attention is then turned to constructing the logical foundation of two different systems of mathematics, one assuming that actual infinity is different than potential infinity, and the other that these infinities are the same. This volume will be of interest to academic researchers, students and professionals in the areas of systems science, mathematics, philosophy of mathematics, and philosophy of science. (shrink)

This book expands the landscape of research in mathematics education by analyzing how the body influences mathematical thinking.

How can we deal with the diversity of theories in mathematics education? This was the main question that led the authors of this book to found the Networking Theories Group. Starting from the shared assumption that the existence of different theories is a resource for mathematics education research, the authors have explored the possibilities of interactions between theories, such as contrasting, coordinating, and locally integrating them. The book explains and illustrates what it means to network theories; it presents networking as (...) a challenging but fruitful research practice and shows how the Group dealt with this challenge considering five theoretical approaches, namely the approach of Action, Production, and Communication (APC), the Theory of Didactical Situations (TDS), the Anthropological Theory of the Didactic (ATD), the approach of Abstraction in Context (AiC), and the Theory of Interest-Dense Situations (IDS). A synthetic presentation of each theory and their connections shows how the activity of networking generates questions at the theoretical, methodological and practical levels and how the work on these questions leads to both theoretical and practical progress. The core of the book consists of four new networking case studies which illustrate what exactly can be gained by this approach and what kind of difficulties might arise. (shrink)

Based on extensive research conducted by the authors, Reasoning and Sense Making in the Mathematics Classroom, Pre-K-Grade 2, is designed to help classroom teachers understand, monitor, and guide the development of students' reasoning and sense making about core ideas in elementary school mathematics. It describes and illustrates the nature of these skills using classroom vignettes and actual student work in conjunction with instructional tasks and learning progressions to show how reasoning and sense making develop and how instruction can support students (...) in that development. Students who can make sense of mathematical ideas can apply those ideas in problem solving, even in unfamiliar situations, and can use them as a foundation for future learning. Without them, students are reduced to rote learning, often experiencing frustration and failure. But what do reasoning and sense making during learning and teaching look like? Each chapter of Reasoning and Sense Making in the Mathematics Classroom, Pre-K-Grade 2 explores a different topic that young children encounter in mathematics, demonstrating with actual student work and classroom dialogue how their mathematical knowledge and reasoning ability move through "levels of sophistication" or learning progressions: After opening with a discussion of the nature of reasoning and sense making and their critical importance in developing mathematical thinking, chapter 1 examines how young students attempt to make sense of the concepts of place value and length measurement. Chapter 2 focuses on how early childhood instruction can engage students in mathematical reasoning while helping them construct a rich sense of number and operations. Chapter 3 identifies core algebraic ideas and shows how students can engage with these ideas in ways that not only deepen their understanding of arithmetic but also lays the foundation for the future study of algebra. Children's reasoning and sense making as they decompose and compose geometric shapes--including reasoning about area--is examined in chapter 4. The use of learning progressions to understand students' reasoning and to guide their sense making with appropriate teaching is also discussed. Not just a theoretical discussion, the book also provides specific suggestions for related instructional activities for each topic. Supplementary online resources can be accessed at NCTM's More4U website. Reasoning and Sense Making in the Mathematics Classroom, Pre-K-Grade 2 will be a valuable and practical addition to your professional library. (shrink)

Despite the importance of mathematics in our educational systems little is known about how abstract mathematical thinking emerges. Under the uniting thread of mathematical development, we hope to connect researchers from various backgrounds to provide an integrated view of abstract mathematical cognition. Much progress has been made in the last 20 years on how numeracy is acquired. Experimental psychology has brought to light the fact that numerical cognition stems from spatial cognition. The findings from neuroimaging and single cell recording experiments (...) converge to show that numerical representations take place in the intraparietal sulcus. Further research has demonstrated that supplementary neural networks might be recruited to carry out subtasks; for example, the retrieval of arithmetic facts is done by the angular gyrus. Now that the neural networks in charge of basic mathematical cognition are identified, we can move onto the stage where we seek to understand how these basics skills are used to support the acquisition and use of abstract mathematical concepts. (shrink)

When it’s time for a game change, you need a guide to the new rules. Helping Students Make Sense of the World Using Next Generation Science and Engineering Practices provides a play-by-play understanding of the practices strand of A Framework for K–12 Science Education (Framework) and the Next Generation Science Standards (NGSS). Written in clear, nontechnical language, this book provides a wealth of real-world examples to show you what’s different about practice-centered teaching and learning at all grade levels. The book (...) addresses three important questions: 1. How will engaging students in science and engineering practices help improve science education? 2. What do the eight practices look like in the classroom? 3. How can educators engage students in practices to bring the NGSS to life? Helping Students Make Sense of the World Using Next Generation Science and Engineering Practices was developed for K–12 science teachers, curriculum developers, teacher educators, and administrators. Many of its authors contributed to the Framework’s initial vision and tested their ideas in actual science classrooms. If you want a fresh game plan to help students work together to generate and revise knowledge—not just receive and repeat information—this book is for you. (shrink)

"This book is interesting and well-written. The research methods were explained clearly and conclusions were summarized nicely. It is a relatively quick read at only 130 pages. Anyone who has been told, or who has told others, that mathematicians make better thinkers should read this book." MAA Reviews "The authors particularly attend to protecting positive correlations against the self-selection interpretation, merely that logical minds elect studying more mathematics. Here, one finds a stimulating survey of the systemic difficulties people have with (...) basic syllogisms and deductions." CHOICE connect "The authors particularly attend to protecting positive correlations against the self-selection interpretation, merely that logical minds elect studying more mathematics. Here, one finds a stimulating survey of the systemic difficulties people have with basic syllogisms and deductions." CHOICE Connect For centuries, educational policymakers have believed that studying mathematics is important, in part because it develops general thinking skills that are useful throughout life. This 'Theory of Formal Discipline' (TFD) has been used as a justification for mathematics education globally. Despite this, few empirical studies have directly investigated the issue, and those which have showed mixed results. Does Mathematical Study Develop Logical Thinking? describes a rigorous investigation of the TFD. It reviews the theory's history and prior research on the topic, followed by reports on a series of recent empirical studies. It argues that, contrary to the position held by sceptics, advanced mathematical study does develop certain general thinking skills, however these are much more restricted than those typically claimed by TFD proponents. Perfect for students, researchers and policymakers in education, further education and mathematics, this book provides much needed insight into the theory and practice of the foundations of modern educational policy. (shrink)

If mathematics is the purest form of knowledge, the perfect foundation of all the hard sciences, and a uniquely precise discipline, then how can the human brain, an imperfect and imprecise organ, process mathematical ideas? Is mathematics made up of eternal, universal truths? Or, as some have claimed, could mathematics simply be a human invention, a kind of tool or metaphor? These questions are among the greatest enigmas of science and epistemology, discussed at length by mathematicians, physicians, and philosophers. But, (...) curiously enough, neuroscientists have been absent in the debate, even though it is precisely the field of neuroscience—which studies the brain’s mechanisms for thinking and reasoning—that ought to be at the very center of these discussions. How our Emotions and Bodies are Vital for Abstract Thought explores the unique mechanisms of cooperation between the body, emotions, and the cortex, based on fundamental physical principles. It is these mechanisms that help us to overcome the limitations of our physiology and allow our imperfect, human brains to make transcendent mathematical discoveries. This book is written for anyone who is interested in the nature of abstract thought, including mathematicians, physicists, computer scientists, psychologists, and psychiatrists. (shrink)

What is argumentation? -- Building a classroom culture of argumentation -- Structuring classroom discussions to focus on argumentation -- Infusing all instruction with argumentation -- Argumentation for all students -- Argumentation and the mathematical practices -- Technology in teaching and learning argumentation -- Assessing argumentation and proof -- Conclusion.